id	sid	tid	token	lemma	pos
asir-36063	1	1	applied	apply	VERB
asir-36063	1	2	science	science	NOUN
asir-36063	1	3	and	and	CCONJ
asir-36063	1	4	innovative	innovative	ADJ
asir-36063	1	5	research	research	NOUN
asir-36063	1	6	issn	issn	VERB
asir-36063	1	7	2474	2474	NUM
asir-36063	1	8	-	-	SYM
asir-36063	1	9	4972	4972	NUM
asir-36063	1	10	(	(	PUNCT
asir-36063	1	11	print	print	NOUN
asir-36063	1	12	)	)	PUNCT
asir-36063	1	13	issn	issn	VERB
asir-36063	1	14	2474	2474	NUM
asir-36063	1	15	-	-	SYM
asir-36063	1	16	4980	4980	NUM
asir-36063	1	17	(	(	PUNCT
asir-36063	1	18	online	online	ADJ
asir-36063	1	19	)	)	PUNCT
asir-36063	1	20	vol	vol	NOUN
asir-36063	1	21	.	.	NOUN
asir-36063	1	22	8	8	NUM
asir-36063	1	23	,	,	PUNCT
asir-36063	1	24	no	no	INTJ
asir-36063	1	25	.	.	NOUN
asir-36063	1	26	1	1	NUM
asir-36063	1	27	,	,	PUNCT
asir-36063	1	28	2024	2024	NUM
asir-36063	1	29	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	1	30	63	63	NUM
asir-36063	1	31	original	original	ADJ
asir-36063	1	32	paper	paper	NOUN
asir-36063	1	33	learning	learning	NOUN
asir-36063	1	34	guarantee	guarantee	NOUN
asir-36063	1	35	of	of	ADP
asir-36063	1	36	sdp	sdp	NOUN
asir-36063	1	37	relaxation	relaxation	NOUN
asir-36063	1	38	on	on	ADP
asir-36063	1	39	directed	direct	VERB
asir-36063	1	40	stochastic	stochastic	ADJ
asir-36063	1	41	block	block	NOUN
asir-36063	1	42	models	model	NOUN
asir-36063	1	43	yanlang	yanlang	NOUN
asir-36063	1	44	chen1	chen1	PROPN
asir-36063	1	45	1	1	NUM
asir-36063	1	46	shenzhen	shenzhen	PROPN
asir-36063	1	47	college	college	PROPN
asir-36063	1	48	of	of	ADP
asir-36063	1	49	international	international	ADJ
asir-36063	1	50	education	education	NOUN
asir-36063	1	51	,	,	PUNCT
asir-36063	1	52	shenzhen	shenzhen	PROPN
asir-36063	1	53	,	,	PUNCT
asir-36063	1	54	guangdong	guangdong	PROPN
asir-36063	1	55	,	,	PUNCT
asir-36063	1	56	518040	518040	NUM
asir-36063	1	57	,	,	PUNCT
asir-36063	1	58	china	china	PROPN
asir-36063	1	59	received	receive	VERB
asir-36063	1	60	:	:	PUNCT
asir-36063	1	61	december	december	PROPN
asir-36063	1	62	18	18	NUM
asir-36063	1	63	,	,	PUNCT
asir-36063	1	64	2023	2023	NUM
asir-36063	1	65	accepted	accept	VERB
asir-36063	1	66	:	:	PUNCT
asir-36063	1	67	january	january	PROPN
asir-36063	1	68	27	27	NUM
asir-36063	1	69	,	,	PUNCT
asir-36063	1	70	2024	2024	NUM
asir-36063	1	71	online	online	ADV
asir-36063	1	72	published	publish	VERB
asir-36063	1	73	:	:	PUNCT
asir-36063	1	74	february	february	PROPN
asir-36063	1	75	2	2	NUM
asir-36063	1	76	,	,	PUNCT
asir-36063	1	77	2024	2024	NUM
asir-36063	1	78	doi:10.22158	doi:10.22158	NOUN
asir-36063	1	79	/	/	SYM
asir-36063	1	80	asir.v8n1p63	asir.v8n1p63	PROPN
asir-36063	1	81	url	url	PROPN
asir-36063	1	82	:	:	PUNCT
asir-36063	1	83	http://doi.org/10.22158/asir.v8n1p63	http://doi.org/10.22158/asir.v8n1p63	PROPN
asir-36063	1	84	abstract	abstract	ADJ
asir-36063	1	85	community	community	NOUN
asir-36063	1	86	detection	detection	NOUN
asir-36063	1	87	in	in	ADP
asir-36063	1	88	networks	network	NOUN
asir-36063	1	89	has	have	AUX
asir-36063	1	90	been	be	AUX
asir-36063	1	91	a	a	DET
asir-36063	1	92	focal	focal	ADJ
asir-36063	1	93	point	point	NOUN
asir-36063	1	94	in	in	ADP
asir-36063	1	95	various	various	ADJ
asir-36063	1	96	scientific	scientific	ADJ
asir-36063	1	97	domains	domain	NOUN
asir-36063	1	98	,	,	PUNCT
asir-36063	1	99	but	but	CCONJ
asir-36063	1	100	the	the	DET
asir-36063	1	101	study	study	NOUN
asir-36063	1	102	of	of	ADP
asir-36063	1	103	directed	direct	VERB
asir-36063	1	104	networks	network	NOUN
asir-36063	1	105	remains	remain	VERB
asir-36063	1	106	relatively	relatively	ADV
asir-36063	1	107	under	under	ADV
asir-36063	1	108	-	-	PUNCT
asir-36063	1	109	explored	explore	VERB
asir-36063	1	110	despite	despite	SCONJ
asir-36063	1	111	their	their	PRON
asir-36063	1	112	prevalence	prevalence	NOUN
asir-36063	1	113	and	and	CCONJ
asir-36063	1	114	importance	importance	NOUN
asir-36063	1	115	in	in	ADP
asir-36063	1	116	capturing	capture	VERB
asir-36063	1	117	real	real	ADJ
asir-36063	1	118	-	-	PUNCT
asir-36063	1	119	world	world	NOUN
asir-36063	1	120	systems	system	NOUN
asir-36063	1	121	.	.	PUNCT
asir-36063	2	1	this	this	DET
asir-36063	2	2	work	work	NOUN
asir-36063	2	3	addresses	address	VERB
asir-36063	2	4	this	this	DET
asir-36063	2	5	research	research	NOUN
asir-36063	2	6	gap	gap	NOUN
asir-36063	2	7	by	by	ADP
asir-36063	2	8	focusing	focus	VERB
asir-36063	2	9	on	on	ADP
asir-36063	2	10	directed	direct	VERB
asir-36063	2	11	stochastic	stochastic	ADJ
asir-36063	2	12	block	block	NOUN
asir-36063	2	13	models	model	NOUN
asir-36063	2	14	(	(	PUNCT
asir-36063	2	15	dsbms	dsbms	NOUN
asir-36063	2	16	)	)	PUNCT
asir-36063	2	17	,	,	PUNCT
asir-36063	2	18	a	a	DET
asir-36063	2	19	natural	natural	ADJ
asir-36063	2	20	extension	extension	NOUN
asir-36063	2	21	of	of	ADP
asir-36063	2	22	traditional	traditional	ADJ
asir-36063	2	23	stochastic	stochastic	ADJ
asir-36063	2	24	block	block	NOUN
asir-36063	2	25	models	model	NOUN
asir-36063	2	26	(	(	PUNCT
asir-36063	2	27	sbms	sbms	ADJ
asir-36063	2	28	)	)	PUNCT
asir-36063	2	29	to	to	ADP
asir-36063	2	30	directed	direct	VERB
asir-36063	2	31	graphs	graph	NOUN
asir-36063	2	32	.	.	PUNCT
asir-36063	3	1	the	the	DET
asir-36063	3	2	inherent	inherent	ADJ
asir-36063	3	3	complexity	complexity	NOUN
asir-36063	3	4	of	of	ADP
asir-36063	3	5	directionality	directionality	NOUN
asir-36063	3	6	in	in	ADP
asir-36063	3	7	dsbms	dsbms	NOUN
asir-36063	3	8	makes	make	VERB
asir-36063	3	9	them	they	PRON
asir-36063	3	10	challenging	challenge	VERB
asir-36063	3	11	to	to	PART
asir-36063	3	12	analyze	analyze	VERB
asir-36063	3	13	,	,	PUNCT
asir-36063	3	14	requiring	require	VERB
asir-36063	3	15	new	new	ADJ
asir-36063	3	16	mathematical	mathematical	ADJ
asir-36063	3	17	frameworks	framework	NOUN
asir-36063	3	18	and	and	CCONJ
asir-36063	3	19	computational	computational	ADJ
asir-36063	3	20	approaches	approach	NOUN
asir-36063	3	21	.	.	PUNCT
asir-36063	4	1	we	we	PRON
asir-36063	4	2	introduce	introduce	VERB
asir-36063	4	3	an	an	DET
asir-36063	4	4	augmented	augment	VERB
asir-36063	4	5	matrix	matrix	NOUN
asir-36063	4	6	to	to	PART
asir-36063	4	7	encapsulate	encapsulate	VERB
asir-36063	4	8	the	the	DET
asir-36063	4	9	directional	directional	ADJ
asir-36063	4	10	relationships	relationship	NOUN
asir-36063	4	11	within	within	ADP
asir-36063	4	12	these	these	DET
asir-36063	4	13	networks	network	NOUN
asir-36063	4	14	,	,	PUNCT
asir-36063	4	15	providing	provide	VERB
asir-36063	4	16	a	a	DET
asir-36063	4	17	nuanced	nuanced	ADJ
asir-36063	4	18	perspective	perspective	NOUN
asir-36063	4	19	for	for	ADP
asir-36063	4	20	further	further	ADJ
asir-36063	4	21	analysis	analysis	NOUN
asir-36063	4	22	.	.	PUNCT
asir-36063	5	1	in	in	ADP
asir-36063	5	2	this	this	DET
asir-36063	5	3	work	work	NOUN
asir-36063	5	4	,	,	PUNCT
asir-36063	5	5	we	we	PRON
asir-36063	5	6	prove	prove	VERB
asir-36063	5	7	the	the	DET
asir-36063	5	8	information	information	NOUN
asir-36063	5	9	-	-	PUNCT
asir-36063	5	10	theoretical	theoretical	ADJ
asir-36063	5	11	threshold	threshold	NOUN
asir-36063	5	12	for	for	ADP
asir-36063	5	13	exact	exact	ADJ
asir-36063	5	14	recovery	recovery	NOUN
asir-36063	5	15	in	in	ADP
asir-36063	5	16	the	the	DET
asir-36063	5	17	dsbms	dsbms	NOUN
asir-36063	5	18	and	and	CCONJ
asir-36063	5	19	propose	propose	VERB
asir-36063	5	20	an	an	DET
asir-36063	5	21	sdp	sdp	NOUN
asir-36063	5	22	relaxation	relaxation	NOUN
asir-36063	5	23	that	that	PRON
asir-36063	5	24	can	can	AUX
asir-36063	5	25	achieve	achieve	VERB
asir-36063	5	26	this	this	DET
asir-36063	5	27	threshold	threshold	NOUN
asir-36063	5	28	,	,	PUNCT
asir-36063	5	29	thereby	thereby	ADV
asir-36063	5	30	contributing	contribute	VERB
asir-36063	5	31	to	to	ADP
asir-36063	5	32	the	the	DET
asir-36063	5	33	theoretical	theoretical	ADJ
asir-36063	5	34	understanding	understanding	NOUN
asir-36063	5	35	of	of	ADP
asir-36063	5	36	community	community	NOUN
asir-36063	5	37	detection	detection	NOUN
asir-36063	5	38	in	in	ADP
asir-36063	5	39	the	the	DET
asir-36063	5	40	realm	realm	NOUN
asir-36063	5	41	of	of	ADP
asir-36063	5	42	directed	direct	VERB
asir-36063	5	43	graphs	graph	NOUN
asir-36063	5	44	.	.	PUNCT
asir-36063	6	1	keywords	keyword	NOUN
asir-36063	6	2	directed	direct	VERB
asir-36063	6	3	stochastic	stochastic	ADJ
asir-36063	6	4	block	block	NOUN
asir-36063	6	5	models	model	NOUN
asir-36063	6	6	,	,	PUNCT
asir-36063	6	7	semi	semi	ADJ
asir-36063	6	8	-	-	ADJ
asir-36063	6	9	definite	definite	ADJ
asir-36063	6	10	programming	programming	NOUN
asir-36063	6	11	(	(	PUNCT
asir-36063	6	12	sdp	sdp	NOUN
asir-36063	6	13	)	)	PUNCT
asir-36063	6	14	relaxation	relaxation	NOUN
asir-36063	6	15	,	,	PUNCT
asir-36063	6	16	clustering	clustering	NOUN
asir-36063	6	17	,	,	PUNCT
asir-36063	6	18	random	random	ADJ
asir-36063	6	19	graph	graph	NOUN
asir-36063	6	20	,	,	PUNCT
asir-36063	6	21	unsupervised	unsupervised	ADJ
asir-36063	6	22	learning	learning	NOUN
asir-36063	6	23	,	,	PUNCT
asir-36063	6	24	community	community	NOUN
asir-36063	6	25	detection	detection	NOUN
asir-36063	6	26	,	,	PUNCT
asir-36063	6	27	directed	direct	VERB
asir-36063	6	28	graphs	graph	NOUN
asir-36063	6	29	1	1	NUM
asir-36063	6	30	.	.	PUNCT
asir-36063	7	1	introduction	introduction	NOUN
asir-36063	7	2	community	community	NOUN
asir-36063	7	3	detection	detection	NOUN
asir-36063	7	4	and	and	CCONJ
asir-36063	7	5	clustering	clustering	NOUN
asir-36063	7	6	are	be	AUX
asir-36063	7	7	pivotal	pivotal	ADJ
asir-36063	7	8	challenges	challenge	NOUN
asir-36063	7	9	in	in	ADP
asir-36063	7	10	an	an	DET
asir-36063	7	11	array	array	NOUN
asir-36063	7	12	of	of	ADP
asir-36063	7	13	disciplines	discipline	NOUN
asir-36063	7	14	,	,	PUNCT
asir-36063	7	15	ranging	range	VERB
asir-36063	7	16	from	from	ADP
asir-36063	7	17	machine	machine	NOUN
asir-36063	7	18	learning	learning	NOUN
asir-36063	7	19	and	and	CCONJ
asir-36063	7	20	data	datum	NOUN
asir-36063	7	21	science	science	NOUN
asir-36063	7	22	to	to	ADP
asir-36063	7	23	the	the	DET
asir-36063	7	24	study	study	NOUN
asir-36063	7	25	of	of	ADP
asir-36063	7	26	complex	complex	ADJ
asir-36063	7	27	networks	network	NOUN
asir-36063	7	28	(	(	PUNCT
asir-36063	7	29	girvan	girvan	PROPN
asir-36063	7	30	&	&	CCONJ
asir-36063	7	31	newman	newman	PROPN
asir-36063	7	32	,	,	PUNCT
asir-36063	7	33	2002	2002	NUM
asir-36063	7	34	;	;	PUNCT
asir-36063	7	35	newman	newman	PROPN
asir-36063	7	36	,	,	PUNCT
asir-36063	7	37	2003	2003	NUM
asir-36063	7	38	)	)	PUNCT
asir-36063	7	39	.	.	PUNCT
asir-36063	8	1	one	one	NUM
asir-36063	8	2	of	of	ADP
asir-36063	8	3	the	the	DET
asir-36063	8	4	most	most	ADV
asir-36063	8	5	striking	striking	ADJ
asir-36063	8	6	features	feature	NOUN
asir-36063	8	7	of	of	ADP
asir-36063	8	8	any	any	DET
asir-36063	8	9	network	network	NOUN
asir-36063	8	10	is	be	AUX
asir-36063	8	11	its	its	PRON
asir-36063	8	12	unique	unique	ADJ
asir-36063	8	13	structure	structure	NOUN
asir-36063	8	14	,	,	PUNCT
asir-36063	8	15	which	which	PRON
asir-36063	8	16	becomes	become	VERB
asir-36063	8	17	evident	evident	ADJ
asir-36063	8	18	through	through	ADP
asir-36063	8	19	the	the	DET
asir-36063	8	20	patterns	pattern	NOUN
asir-36063	8	21	of	of	ADP
asir-36063	8	22	interaction	interaction	NOUN
asir-36063	8	23	among	among	ADP
asir-36063	8	24	its	its	PRON
asir-36063	8	25	vertices	vertex	NOUN
asir-36063	8	26	.	.	PUNCT
asir-36063	9	1	for	for	ADP
asir-36063	9	2	instance	instance	NOUN
asir-36063	9	3	,	,	PUNCT
asir-36063	9	4	certain	certain	ADJ
asir-36063	9	5	subsets	subset	NOUN
asir-36063	9	6	of	of	ADP
asir-36063	9	7	vertices	vertex	NOUN
asir-36063	9	8	in	in	ADP
asir-36063	9	9	a	a	DET
asir-36063	9	10	vast	vast	ADJ
asir-36063	9	11	network	network	NOUN
asir-36063	9	12	are	be	AUX
asir-36063	9	13	tightly	tightly	ADV
asir-36063	9	14	interlinked	interlink	VERB
asir-36063	9	15	,	,	PUNCT
asir-36063	9	16	while	while	SCONJ
asir-36063	9	17	their	their	PRON
asir-36063	9	18	connections	connection	NOUN
asir-36063	9	19	to	to	ADP
asir-36063	9	20	vertices	vertex	NOUN
asir-36063	9	21	outside	outside	ADP
asir-36063	9	22	this	this	DET
asir-36063	9	23	cluster	cluster	NOUN
asir-36063	9	24	are	be	AUX
asir-36063	9	25	notably	notably	ADV
asir-36063	9	26	sparse	sparse	ADJ
asir-36063	9	27	.	.	PUNCT
asir-36063	10	1	while	while	SCONJ
asir-36063	10	2	substantial	substantial	ADJ
asir-36063	10	3	research	research	NOUN
asir-36063	10	4	has	have	AUX
asir-36063	10	5	focused	focus	VERB
asir-36063	10	6	on	on	ADP
asir-36063	10	7	undirected	undirected	ADJ
asir-36063	10	8	networks	network	NOUN
asir-36063	10	9	such	such	ADJ
asir-36063	10	10	as	as	ADP
asir-36063	10	11	geographical	geographical	ADJ
asir-36063	10	12	maps	map	NOUN
asir-36063	10	13	,	,	PUNCT
asir-36063	10	14	friendship	friendship	NOUN
asir-36063	10	15	circles	circle	NOUN
asir-36063	10	16	,	,	PUNCT
asir-36063	10	17	and	and	CCONJ
asir-36063	10	18	familial	familial	ADJ
asir-36063	10	19	connections	connection	NOUN
asir-36063	10	20	there	there	PRON
asir-36063	10	21	is	be	VERB
asir-36063	10	22	a	a	DET
asir-36063	10	23	compelling	compelling	ADJ
asir-36063	10	24	yet	yet	CCONJ
asir-36063	10	25	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	10	26	applied	apply	VERB
asir-36063	10	27	science	science	NOUN
asir-36063	10	28	and	and	CCONJ
asir-36063	10	29	innovative	innovative	ADJ
asir-36063	10	30	research	research	NOUN
asir-36063	10	31	vol	vol	NOUN
asir-36063	10	32	.	.	PROPN
asir-36063	10	33	8	8	NUM
asir-36063	10	34	,	,	PUNCT
asir-36063	10	35	no	no	INTJ
asir-36063	10	36	.	.	NOUN
asir-36063	10	37	1	1	NUM
asir-36063	10	38	,	,	PUNCT
asir-36063	10	39	2024	2024	NUM
asir-36063	10	40	64	64	NUM
asir-36063	10	41	published	publish	VERB
asir-36063	10	42	by	by	ADP
asir-36063	10	43	scholink	scholink	PROPN
asir-36063	10	44	inc	inc	PROPN
asir-36063	10	45	.	.	PROPN
asir-36063	10	46	underexplored	underexplored	ADJ
asir-36063	10	47	frontier	frontier	NOUN
asir-36063	10	48	in	in	ADP
asir-36063	10	49	the	the	DET
asir-36063	10	50	realm	realm	NOUN
asir-36063	10	51	of	of	ADP
asir-36063	10	52	directed	direct	VERB
asir-36063	10	53	networks	network	NOUN
asir-36063	10	54	.	.	PUNCT
asir-36063	11	1	directed	direct	VERB
asir-36063	11	2	networks	network	NOUN
asir-36063	11	3	,	,	PUNCT
asir-36063	11	4	evident	evident	ADJ
asir-36063	11	5	in	in	ADP
asir-36063	11	6	phenomena	phenomenon	NOUN
asir-36063	11	7	like	like	ADP
asir-36063	11	8	social	social	ADJ
asir-36063	11	9	media	medium	NOUN
asir-36063	11	10	interactions	interaction	NOUN
asir-36063	11	11	,	,	PUNCT
asir-36063	11	12	web	web	NOUN
asir-36063	11	13	page	page	NOUN
asir-36063	11	14	hyperlinks	hyperlink	NOUN
asir-36063	11	15	,	,	PUNCT
asir-36063	11	16	and	and	CCONJ
asir-36063	11	17	aviation	aviation	NOUN
asir-36063	11	18	routes	route	NOUN
asir-36063	11	19	,	,	PUNCT
asir-36063	11	20	more	more	ADV
asir-36063	11	21	closely	closely	ADV
asir-36063	11	22	mimic	mimic	VERB
asir-36063	11	23	the	the	DET
asir-36063	11	24	intricacies	intricacy	NOUN
asir-36063	11	25	of	of	ADP
asir-36063	11	26	real	real	ADJ
asir-36063	11	27	-	-	PUNCT
asir-36063	11	28	world	world	NOUN
asir-36063	11	29	systems	system	NOUN
asir-36063	11	30	.	.	PUNCT
asir-36063	12	1	their	their	PRON
asir-36063	12	2	inherent	inherent	ADJ
asir-36063	12	3	directionality	directionality	NOUN
asir-36063	12	4	not	not	PART
asir-36063	12	5	only	only	ADV
asir-36063	12	6	makes	make	VERB
asir-36063	12	7	them	they	PRON
asir-36063	12	8	more	more	ADV
asir-36063	12	9	relevant	relevant	ADJ
asir-36063	12	10	for	for	ADP
asir-36063	12	11	practical	practical	ADJ
asir-36063	12	12	applications	application	NOUN
asir-36063	12	13	but	but	CCONJ
asir-36063	12	14	also	also	ADV
asir-36063	12	15	significantly	significantly	ADV
asir-36063	12	16	more	more	ADV
asir-36063	12	17	challenging	challenging	ADJ
asir-36063	12	18	to	to	PART
asir-36063	12	19	analyze	analyze	VERB
asir-36063	12	20	.	.	PUNCT
asir-36063	13	1	this	this	DET
asir-36063	13	2	very	very	ADJ
asir-36063	13	3	novelty	novelty	NOUN
asir-36063	13	4	and	and	CCONJ
asir-36063	13	5	complexity	complexity	NOUN
asir-36063	13	6	of	of	ADP
asir-36063	13	7	directed	direct	VERB
asir-36063	13	8	networks	network	NOUN
asir-36063	13	9	serve	serve	VERB
asir-36063	13	10	as	as	ADP
asir-36063	13	11	the	the	DET
asir-36063	13	12	driving	drive	VERB
asir-36063	13	13	force	force	NOUN
asir-36063	13	14	behind	behind	ADP
asir-36063	13	15	this	this	DET
asir-36063	13	16	thesis	thesis	NOUN
asir-36063	13	17	.	.	PUNCT
asir-36063	14	1	stochastic	stochastic	ADJ
asir-36063	14	2	block	block	NOUN
asir-36063	14	3	models	model	NOUN
asir-36063	14	4	(	(	PUNCT
asir-36063	14	5	sbms	sbms	ADJ
asir-36063	14	6	)	)	PUNCT
asir-36063	14	7	(	(	PUNCT
asir-36063	14	8	emmanuel	emmanuel	PROPN
asir-36063	14	9	,	,	PUNCT
asir-36063	14	10	afonso	afonso	PROPN
asir-36063	14	11	,	,	PUNCT
asir-36063	14	12	&	&	CCONJ
asir-36063	14	13	georgina	georgina	PROPN
asir-36063	14	14	,	,	PUNCT
asir-36063	14	15	2014	2014	NUM
asir-36063	14	16	)	)	PUNCT
asir-36063	14	17	have	have	AUX
asir-36063	14	18	traditionally	traditionally	ADV
asir-36063	14	19	been	be	AUX
asir-36063	14	20	employed	employ	VERB
asir-36063	14	21	to	to	PART
asir-36063	14	22	examine	examine	VERB
asir-36063	14	23	random	random	ADJ
asir-36063	14	24	block	block	NOUN
asir-36063	14	25	structures	structure	NOUN
asir-36063	14	26	,	,	PUNCT
asir-36063	14	27	originally	originally	ADV
asir-36063	14	28	formulated	formulate	VERB
asir-36063	14	29	to	to	PART
asir-36063	14	30	scrutinize	scrutinize	VERB
asir-36063	14	31	social	social	ADJ
asir-36063	14	32	networks	network	NOUN
asir-36063	14	33	.	.	PUNCT
asir-36063	15	1	this	this	DET
asir-36063	15	2	model	model	NOUN
asir-36063	15	3	serves	serve	VERB
asir-36063	15	4	as	as	ADP
asir-36063	15	5	a	a	DET
asir-36063	15	6	powerful	powerful	ADJ
asir-36063	15	7	benchmark	benchmark	NOUN
asir-36063	15	8	for	for	ADP
asir-36063	15	9	assessing	assess	VERB
asir-36063	15	10	the	the	DET
asir-36063	15	11	performance	performance	NOUN
asir-36063	15	12	of	of	ADP
asir-36063	15	13	clustering	clustering	ADJ
asir-36063	15	14	algorithms	algorithm	NOUN
asir-36063	15	15	.	.	PUNCT
asir-36063	16	1	however	however	ADV
asir-36063	16	2	,	,	PUNCT
asir-36063	16	3	its	its	PRON
asir-36063	16	4	main	main	ADJ
asir-36063	16	5	limitation	limitation	NOUN
asir-36063	16	6	lies	lie	VERB
asir-36063	16	7	in	in	ADP
asir-36063	16	8	its	its	PRON
asir-36063	16	9	oversimplification	oversimplification	NOUN
asir-36063	16	10	of	of	ADP
asir-36063	16	11	real	real	ADJ
asir-36063	16	12	-	-	PUNCT
asir-36063	16	13	world	world	NOUN
asir-36063	16	14	networks	network	NOUN
asir-36063	16	15	,	,	PUNCT
asir-36063	16	16	particularly	particularly	ADV
asir-36063	16	17	due	due	ADP
asir-36063	16	18	to	to	ADP
asir-36063	16	19	its	its	PRON
asir-36063	16	20	strong	strong	ADJ
asir-36063	16	21	homogeneity	homogeneity	NOUN
asir-36063	16	22	and	and	CCONJ
asir-36063	16	23	lack	lack	NOUN
asir-36063	16	24	of	of	ADP
asir-36063	16	25	community	community	NOUN
asir-36063	16	26	structure	structure	NOUN
asir-36063	16	27	.	.	PUNCT
asir-36063	17	1	moreover	moreover	ADV
asir-36063	17	2	,	,	PUNCT
asir-36063	17	3	the	the	DET
asir-36063	17	4	burgeoning	burgeon	VERB
asir-36063	17	5	research	research	NOUN
asir-36063	17	6	in	in	ADP
asir-36063	17	7	this	this	DET
asir-36063	17	8	area	area	NOUN
asir-36063	17	9	has	have	AUX
asir-36063	17	10	disproportionately	disproportionately	ADV
asir-36063	17	11	focused	focus	VERB
asir-36063	17	12	on	on	ADP
asir-36063	17	13	undirected	undirected	ADJ
asir-36063	17	14	sbms	sbms	NOUN
asir-36063	17	15	(	(	PUNCT
asir-36063	17	16	andrea	andrea	PROPN
asir-36063	17	17	,	,	PUNCT
asir-36063	17	18	santo	santo	PROPN
asir-36063	17	19	,	,	PUNCT
asir-36063	17	20	&	&	CCONJ
asir-36063	17	21	filippo	filippo	PROPN
asir-36063	17	22	,	,	PUNCT
asir-36063	17	23	2008	2008	NUM
asir-36063	17	24	;	;	PUNCT
asir-36063	17	25	santo	santo	PROPN
asir-36063	17	26	,	,	PUNCT
asir-36063	17	27	2010	2010	NUM
asir-36063	17	28	)	)	PUNCT
asir-36063	17	29	,	,	PUNCT
asir-36063	17	30	thus	thus	ADV
asir-36063	17	31	leaving	leave	VERB
asir-36063	17	32	a	a	DET
asir-36063	17	33	crucial	crucial	ADJ
asir-36063	17	34	gap	gap	NOUN
asir-36063	17	35	in	in	ADP
asir-36063	17	36	our	our	PRON
asir-36063	17	37	understanding	understanding	NOUN
asir-36063	17	38	of	of	ADP
asir-36063	17	39	directed	direct	VERB
asir-36063	17	40	stochastic	stochastic	ADJ
asir-36063	17	41	block	block	NOUN
asir-36063	17	42	models	model	NOUN
asir-36063	17	43	(	(	PUNCT
asir-36063	17	44	dsbms	dsbms	NOUN
asir-36063	17	45	)	)	PUNCT
asir-36063	17	46	.	.	PUNCT
asir-36063	18	1	the	the	DET
asir-36063	18	2	challenge	challenge	NOUN
asir-36063	18	3	in	in	ADP
asir-36063	18	4	dsbms	dsbms	NOUN
asir-36063	18	5	is	be	AUX
asir-36063	18	6	not	not	PART
asir-36063	18	7	merely	merely	ADV
asir-36063	18	8	a	a	DET
asir-36063	18	9	replication	replication	NOUN
asir-36063	18	10	of	of	ADP
asir-36063	18	11	its	its	PRON
asir-36063	18	12	undirected	undirected	ADJ
asir-36063	18	13	counterparts	counterpart	NOUN
asir-36063	18	14	;	;	PUNCT
asir-36063	18	15	it	it	PRON
asir-36063	18	16	is	be	AUX
asir-36063	18	17	profoundly	profoundly	ADV
asir-36063	18	18	exacerbated	exacerbate	VERB
asir-36063	18	19	by	by	ADP
asir-36063	18	20	the	the	DET
asir-36063	18	21	added	add	VERB
asir-36063	18	22	complexity	complexity	NOUN
asir-36063	18	23	of	of	ADP
asir-36063	18	24	directionality	directionality	NOUN
asir-36063	18	25	.	.	PUNCT
asir-36063	19	1	in	in	ADP
asir-36063	19	2	light	light	NOUN
asir-36063	19	3	of	of	ADP
asir-36063	19	4	this	this	PRON
asir-36063	19	5	,	,	PUNCT
asir-36063	19	6	we	we	PRON
asir-36063	19	7	leverage	leverage	VERB
asir-36063	19	8	an	an	DET
asir-36063	19	9	augmented	augment	VERB
asir-36063	19	10	matrix	matrix	NOUN
asir-36063	19	11	to	to	PART
asir-36063	19	12	encapsulate	encapsulate	VERB
asir-36063	19	13	these	these	DET
asir-36063	19	14	directional	directional	ADJ
asir-36063	19	15	relationships	relationship	NOUN
asir-36063	19	16	,	,	PUNCT
asir-36063	19	17	providing	provide	VERB
asir-36063	19	18	a	a	DET
asir-36063	19	19	nuanced	nuanced	ADJ
asir-36063	19	20	mathematical	mathematical	ADJ
asir-36063	19	21	framework	framework	NOUN
asir-36063	19	22	to	to	PART
asir-36063	19	23	navigate	navigate	VERB
asir-36063	19	24	this	this	DET
asir-36063	19	25	intricate	intricate	ADJ
asir-36063	19	26	landscape	landscape	NOUN
asir-36063	19	27	.	.	PUNCT
asir-36063	20	1	dsbms	dsbm	NOUN
asir-36063	20	2	also	also	ADV
asir-36063	20	3	possess	possess	VERB
asir-36063	20	4	desirable	desirable	ADJ
asir-36063	20	5	consistency	consistency	NOUN
asir-36063	20	6	properties	property	NOUN
asir-36063	20	7	similar	similar	ADJ
asir-36063	20	8	to	to	ADP
asir-36063	20	9	undirected	undirected	ADJ
asir-36063	20	10	sbms	sbms	NOUN
asir-36063	20	11	,	,	PUNCT
asir-36063	20	12	but	but	CCONJ
asir-36063	20	13	obtaining	obtain	VERB
asir-36063	20	14	exact	exact	ADJ
asir-36063	20	15	parameter	parameter	NOUN
asir-36063	20	16	estimates	estimate	NOUN
asir-36063	20	17	in	in	ADP
asir-36063	20	18	both	both	PRON
asir-36063	20	19	is	be	AUX
asir-36063	20	20	generally	generally	ADV
asir-36063	20	21	an	an	DET
asir-36063	20	22	np	np	ADV
asir-36063	20	23	-	-	PUNCT
asir-36063	20	24	hard	hard	ADJ
asir-36063	20	25	problem	problem	NOUN
asir-36063	20	26	.	.	PUNCT
asir-36063	21	1	inspired	inspire	VERB
asir-36063	21	2	by	by	ADP
asir-36063	21	3	semi	semi	ADJ
asir-36063	21	4	-	-	ADJ
asir-36063	21	5	definite	definite	ADJ
asir-36063	21	6	programming	programming	NOUN
asir-36063	21	7	(	(	PUNCT
asir-36063	21	8	sdp	sdp	NOUN
asir-36063	21	9	)	)	PUNCT
asir-36063	21	10	relaxation	relaxation	NOUN
asir-36063	21	11	techniques	technique	NOUN
asir-36063	21	12	(	(	PUNCT
asir-36063	21	13	afonso	afonso	PROPN
asir-36063	21	14	,	,	PUNCT
asir-36063	21	15	2015	2015	NUM
asir-36063	21	16	)	)	PUNCT
asir-36063	21	17	,	,	PUNCT
asir-36063	21	18	our	our	PRON
asir-36063	21	19	work	work	NOUN
asir-36063	21	20	aims	aim	VERB
asir-36063	21	21	to	to	PART
asir-36063	21	22	bypass	bypass	VERB
asir-36063	21	23	this	this	DET
asir-36063	21	24	computational	computational	ADJ
asir-36063	21	25	bottleneck	bottleneck	NOUN
asir-36063	21	26	.	.	PUNCT
asir-36063	22	1	we	we	PRON
asir-36063	22	2	offer	offer	VERB
asir-36063	22	3	a	a	DET
asir-36063	22	4	pioneering	pioneering	ADJ
asir-36063	22	5	semi	semi	ADJ
asir-36063	22	6	-	-	ADJ
asir-36063	22	7	definite	definite	ADJ
asir-36063	22	8	relaxation	relaxation	NOUN
asir-36063	22	9	approach	approach	NOUN
asir-36063	22	10	to	to	ADP
asir-36063	22	11	discern	discern	ADJ
asir-36063	22	12	clustering	clustering	ADJ
asir-36063	22	13	thresholds	threshold	NOUN
asir-36063	22	14	in	in	ADP
asir-36063	22	15	dsbms	dsbms	NOUN
asir-36063	22	16	,	,	PUNCT
asir-36063	22	17	thereby	thereby	ADV
asir-36063	22	18	overcoming	overcome	VERB
asir-36063	22	19	the	the	DET
asir-36063	22	20	inherent	inherent	ADJ
asir-36063	22	21	np	np	NOUN
asir-36063	22	22	-	-	PUNCT
asir-36063	22	23	hardness	hardness	NOUN
asir-36063	22	24	.	.	PUNCT
asir-36063	23	1	in	in	ADP
asir-36063	23	2	this	this	DET
asir-36063	23	3	work	work	NOUN
asir-36063	23	4	,	,	PUNCT
asir-36063	23	5	we	we	PRON
asir-36063	23	6	prove	prove	VERB
asir-36063	23	7	the	the	DET
asir-36063	23	8	information	information	NOUN
asir-36063	23	9	-	-	PUNCT
asir-36063	23	10	theoretical	theoretical	ADJ
asir-36063	23	11	threshold	threshold	NOUN
asir-36063	23	12	for	for	ADP
asir-36063	23	13	exact	exact	ADJ
asir-36063	23	14	recovery	recovery	NOUN
asir-36063	23	15	in	in	ADP
asir-36063	23	16	the	the	DET
asir-36063	23	17	directed	direct	VERB
asir-36063	23	18	stochastic	stochastic	ADJ
asir-36063	23	19	block	block	NOUN
asir-36063	23	20	models	model	NOUN
asir-36063	23	21	and	and	CCONJ
asir-36063	23	22	propose	propose	VERB
asir-36063	23	23	an	an	DET
asir-36063	23	24	sdp	sdp	NOUN
asir-36063	23	25	relaxation	relaxation	NOUN
asir-36063	23	26	that	that	PRON
asir-36063	23	27	can	can	AUX
asir-36063	23	28	achieve	achieve	VERB
asir-36063	23	29	the	the	DET
asir-36063	23	30	threshold	threshold	NOUN
asir-36063	23	31	,	,	PUNCT
asir-36063	23	32	which	which	PRON
asir-36063	23	33	fills	fill	VERB
asir-36063	23	34	the	the	DET
asir-36063	23	35	gap	gap	NOUN
asir-36063	23	36	of	of	ADP
asir-36063	23	37	the	the	DET
asir-36063	23	38	theoretical	theoretical	ADJ
asir-36063	23	39	understanding	understanding	NOUN
asir-36063	23	40	of	of	ADP
asir-36063	23	41	community	community	NOUN
asir-36063	23	42	detection	detection	NOUN
asir-36063	23	43	in	in	ADP
asir-36063	23	44	the	the	DET
asir-36063	23	45	context	context	NOUN
asir-36063	23	46	of	of	ADP
asir-36063	23	47	directed	direct	VERB
asir-36063	23	48	graphs	graph	NOUN
asir-36063	23	49	.	.	PUNCT
asir-36063	24	1	1.1	1.1	NUM
asir-36063	24	2	notations	notation	NOUN
asir-36063	24	3	let	let	VERB
asir-36063	24	4	𝑨	𝑨	PROPN
asir-36063	24	5	∈	∈	PRON
asir-36063	24	6	ℂ𝑛×𝑚	ℂ𝑛×𝑚	VERB
asir-36063	24	7	be	be	AUX
asir-36063	24	8	a	a	DET
asir-36063	24	9	complex	complex	ADJ
asir-36063	24	10	matrix	matrix	NOUN
asir-36063	24	11	and	and	CCONJ
asir-36063	24	12	denote	denote	VERB
asir-36063	24	13	its	its	PRON
asir-36063	24	14	(	(	PUNCT
asir-36063	24	15	𝑖	𝑖	NOUN
asir-36063	24	16	,	,	PUNCT
asir-36063	24	17	𝑗)-entry	𝑗)-entry	NOUN
asir-36063	24	18	by	by	ADP
asir-36063	24	19	𝐴𝑖𝑗.	𝐴𝑖𝑗.	PROPN
asir-36063	24	20	we	we	PRON
asir-36063	24	21	denote	denote	VERB
asir-36063	24	22	its	its	PRON
asir-36063	24	23	transpose	transpose	NOUN
asir-36063	24	24	and	and	CCONJ
asir-36063	24	25	conjugate	conjugate	ADJ
asir-36063	24	26	transpose	transpose	NOUN
asir-36063	24	27	as	as	ADP
asir-36063	24	28	𝑨⊤	𝑨⊤	NUM
asir-36063	24	29	and	and	CCONJ
asir-36063	24	30	𝑨𝐻	𝑨𝐻	PROPN
asir-36063	24	31	respectively	respectively	ADV
asir-36063	24	32	.	.	PUNCT
asir-36063	25	1	the	the	DET
asir-36063	25	2	ℓ2	ℓ2	ADJ
asir-36063	25	3	-norm	-norm	NOUN
asir-36063	25	4	of	of	ADP
asir-36063	25	5	a	a	DET
asir-36063	25	6	vector	vector	NOUN
asir-36063	25	7	𝒗	𝒗	PROPN
asir-36063	25	8	is	be	AUX
asir-36063	25	9	denoted	denote	VERB
asir-36063	25	10	by	by	ADP
asir-36063	25	11	∥	∥	PROPN
asir-36063	25	12	𝒗	𝒗	PROPN
asir-36063	25	13	∥=	∥=	NOUN
asir-36063	25	14	√𝒗𝐻𝒗	√𝒗𝐻𝒗	NOUN
asir-36063	25	15	=	=	SYM
asir-36063	25	16	√∑𝑗=1	√∑𝑗=1	NOUN
asir-36063	25	17	𝑛	𝑛	PRON
asir-36063	25	18	 	 	SPACE
asir-36063	25	19	|𝑣𝑗|	|𝑣𝑗|	PROPN
asir-36063	25	20	2	2	NUM
asir-36063	25	21	and	and	CCONJ
asir-36063	25	22	its	its	PRON
asir-36063	25	23	ℓ∞	ℓ∞	PROPN
asir-36063	25	24	norm	norm	NOUN
asir-36063	25	25	is	be	AUX
asir-36063	25	26	denoted	denote	VERB
asir-36063	25	27	by	by	ADP
asir-36063	25	28	∥	∥	PROPN
asir-36063	25	29	𝒗	𝒗	PROPN
asir-36063	25	30	∥∞=	∥∞=	VERB
asir-36063	25	31	max1≤𝑘≤𝑛	max1≤𝑘≤𝑛	ADJ
asir-36063	25	32	 	 	SPACE
asir-36063	25	33	|𝑣𝑘|	|𝑣𝑘|	NOUN
asir-36063	25	34	,	,	PUNCT
asir-36063	25	35	where	where	SCONJ
asir-36063	25	36	𝑣𝑘	𝑣𝑘	PRON
asir-36063	25	37	is	be	AUX
asir-36063	25	38	𝒗	𝒗	PROPN
asir-36063	25	39	's	's	PART
asir-36063	25	40	𝑘-th	𝑘-th	ADJ
asir-36063	25	41	entry	entry	NOUN
asir-36063	25	42	.	.	PUNCT
asir-36063	26	1	the	the	DET
asir-36063	26	2	inner	inner	ADJ
asir-36063	26	3	product	product	NOUN
asir-36063	26	4	between	between	ADP
asir-36063	26	5	two	two	NUM
asir-36063	26	6	complex	complex	ADJ
asir-36063	26	7	vectors	vector	NOUN
asir-36063	26	8	𝒖	𝒖	NOUN
asir-36063	26	9	and	and	CCONJ
asir-36063	26	10	𝒗	𝒗	PROPN
asir-36063	26	11	is	be	AUX
asir-36063	26	12	defined	define	VERB
asir-36063	26	13	as	as	ADP
asir-36063	26	14	⟨𝒖	⟨𝒖	NOUN
asir-36063	26	15	,	,	PUNCT
asir-36063	26	16	𝒗⟩	𝒗⟩	PUNCT
asir-36063	27	1	=	=	PUNCT
asir-36063	27	2	𝒖𝐻𝒗.	𝒖𝐻𝒗.	PROPN
asir-36063	27	3	for	for	ADP
asir-36063	27	4	two	two	NUM
asir-36063	27	5	vectors	vector	NOUN
asir-36063	27	6	𝒖	𝒖	X
asir-36063	27	7	and	and	CCONJ
asir-36063	27	8	𝒗	𝒗	ADJ
asir-36063	27	9	,	,	PUNCT
asir-36063	27	10	we	we	PRON
asir-36063	27	11	denote	denote	VERB
asir-36063	27	12	𝒖	𝒖	X
asir-36063	27	13	∝	∝	PROPN
asir-36063	27	14	𝒗	𝒗	PROPN
asir-36063	27	15	if	if	SCONJ
asir-36063	27	16	they	they	PRON
asir-36063	27	17	are	be	AUX
asir-36063	27	18	parallel	parallel	ADJ
asir-36063	27	19	.	.	PUNCT
asir-36063	28	1	we	we	PRON
asir-36063	28	2	denote	denote	VERB
asir-36063	28	3	the	the	DET
asir-36063	28	4	operator	operator	NOUN
asir-36063	28	5	2	2	NUM
asir-36063	28	6	-	-	PUNCT
asir-36063	28	7	norm	norm	NOUN
asir-36063	28	8	of	of	ADP
asir-36063	28	9	𝑨	𝑨	NOUN
asir-36063	28	10	as	as	ADP
asir-36063	28	11	∥	∥	NUM
asir-36063	28	12	𝑨	𝑨	NOUN
asir-36063	28	13	∥	∥	PUNCT
asir-36063	28	14	which	which	PRON
asir-36063	28	15	is	be	AUX
asir-36063	28	16	the	the	DET
asir-36063	28	17	largest	large	ADJ
asir-36063	28	18	singular	singular	ADJ
asir-36063	28	19	value	value	NOUN
asir-36063	28	20	of	of	ADP
asir-36063	28	21	𝑨.	𝑨.	PROPN
asir-36063	28	22	we	we	PRON
asir-36063	28	23	denote	denote	VERB
asir-36063	28	24	the	the	DET
asir-36063	28	25	all	all	ADV
asir-36063	28	26	-	-	PUNCT
asir-36063	28	27	one	one	NUM
asir-36063	28	28	vector	vector	NOUN
asir-36063	28	29	in	in	ADP
asir-36063	28	30	ℝ𝑛	ℝ𝑛	PROPN
asir-36063	28	31	as	as	ADP
asir-36063	28	32	𝟏𝑛	𝟏𝑛	NOUN
asir-36063	28	33	and	and	CCONJ
asir-36063	28	34	the	the	DET
asir-36063	28	35	all	all	ADV
asir-36063	28	36	-	-	PUNCT
asir-36063	28	37	one	one	NUM
asir-36063	28	38	matrix	matrix	NOUN
asir-36063	28	39	in	in	ADP
asir-36063	28	40	ℝ𝑛×𝑛	ℝ𝑛×𝑛	PROPN
asir-36063	28	41	as	as	ADP
asir-36063	28	42	𝑱𝑛.	𝑱𝑛.	PROPN
asir-36063	28	43	for	for	ADP
asir-36063	28	44	𝑨	𝑨	PROPN
asir-36063	28	45	∈	∈	PROPN
asir-36063	28	46	ℝ𝑛×𝑛	ℝ𝑛×𝑛	PROPN
asir-36063	28	47	,	,	PUNCT
asir-36063	28	48	if	if	SCONJ
asir-36063	28	49	𝑨	𝑨	PROPN
asir-36063	28	50	is	be	AUX
asir-36063	28	51	symmetric	symmetric	ADJ
asir-36063	28	52	and	and	CCONJ
asir-36063	28	53	all	all	DET
asir-36063	28	54	its	its	PRON
asir-36063	28	55	eigenvalues	eigenvalue	NOUN
asir-36063	28	56	are	be	AUX
asir-36063	28	57	non	non	ADJ
asir-36063	28	58	-	-	ADJ
asir-36063	28	59	negative	negative	ADJ
asir-36063	28	60	,	,	PUNCT
asir-36063	28	61	we	we	PRON
asir-36063	28	62	say	say	VERB
asir-36063	28	63	𝑨	𝑨	PROPN
asir-36063	28	64	is	be	AUX
asir-36063	28	65	positive	positive	ADJ
asir-36063	28	66	semidefinite	semidefinite	NOUN
asir-36063	28	67	,	,	PUNCT
asir-36063	28	68	denoted	denote	VERB
asir-36063	28	69	by	by	ADP
asir-36063	28	70	𝑨	𝑨	PROPN
asir-36063	28	71	≽	≽	PROPN
asir-36063	28	72	0	0	NUM
asir-36063	28	73	.	.	PROPN
asir-36063	28	74	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-36063	28	75	applied	apply	VERB
asir-36063	28	76	science	science	NOUN
asir-36063	28	77	and	and	CCONJ
asir-36063	28	78	innovative	innovative	ADJ
asir-36063	28	79	research	research	NOUN
asir-36063	28	80	vol	vol	NOUN
asir-36063	28	81	.	.	PROPN
asir-36063	29	1	8	8	NUM
asir-36063	29	2	,	,	PUNCT
asir-36063	29	3	no	no	INTJ
asir-36063	29	4	.	.	NOUN
asir-36063	29	5	1	1	NUM
asir-36063	29	6	,	,	PUNCT
asir-36063	29	7	2024	2024	NUM
asir-36063	29	8	65	65	NUM
asir-36063	29	9	published	publish	VERB
asir-36063	29	10	by	by	ADP
asir-36063	29	11	scholink	scholink	PROPN
asir-36063	29	12	inc	inc	PROPN
asir-36063	29	13	.	.	PROPN
asir-36063	29	14	2	2	NUM
asir-36063	29	15	.	.	NUM
asir-36063	29	16	preliminaries	preliminary	NOUN
asir-36063	29	17	in	in	ADP
asir-36063	29	18	this	this	DET
asir-36063	29	19	section	section	NOUN
asir-36063	29	20	,	,	PUNCT
asir-36063	29	21	we	we	PRON
asir-36063	29	22	will	will	AUX
asir-36063	29	23	introduce	introduce	VERB
asir-36063	29	24	the	the	DET
asir-36063	29	25	problem	problem	NOUN
asir-36063	29	26	settings	setting	NOUN
asir-36063	29	27	and	and	CCONJ
asir-36063	29	28	the	the	DET
asir-36063	29	29	core	core	NOUN
asir-36063	29	30	definitions	definition	NOUN
asir-36063	29	31	for	for	ADP
asir-36063	29	32	the	the	DET
asir-36063	29	33	paper	paper	NOUN
asir-36063	29	34	.	.	PUNCT
asir-36063	30	1	2.1	2.1	NUM
asir-36063	30	2	directed	direct	VERB
asir-36063	30	3	stochastic	stochastic	ADJ
asir-36063	30	4	block	block	NOUN
asir-36063	30	5	models	model	NOUN
asir-36063	30	6	the	the	DET
asir-36063	30	7	directed	direct	VERB
asir-36063	30	8	stochastic	stochastic	ADJ
asir-36063	30	9	block	block	NOUN
asir-36063	30	10	models	model	NOUN
asir-36063	30	11	(	(	PUNCT
asir-36063	30	12	or	or	CCONJ
asir-36063	30	13	dsbm	dsbm	ADJ
asir-36063	30	14	in	in	ADP
asir-36063	30	15	short	short	ADJ
asir-36063	30	16	)	)	PUNCT
asir-36063	30	17	is	be	AUX
asir-36063	30	18	a	a	DET
asir-36063	30	19	generative	generative	ADJ
asir-36063	30	20	model	model	NOUN
asir-36063	30	21	for	for	ADP
asir-36063	30	22	modeling	model	VERB
asir-36063	30	23	the	the	DET
asir-36063	30	24	community	community	NOUN
asir-36063	30	25	structures	structure	NOUN
asir-36063	30	26	in	in	ADP
asir-36063	30	27	directed	direct	VERB
asir-36063	30	28	networks	network	NOUN
asir-36063	30	29	,	,	PUNCT
asir-36063	30	30	which	which	PRON
asir-36063	30	31	is	be	AUX
asir-36063	30	32	a	a	DET
asir-36063	30	33	benchmark	benchmark	NOUN
asir-36063	30	34	for	for	ADP
asir-36063	30	35	comparing	compare	VERB
asir-36063	30	36	different	different	ADJ
asir-36063	30	37	community	community	NOUN
asir-36063	30	38	detection	detection	NOUN
asir-36063	30	39	methods	method	NOUN
asir-36063	30	40	.	.	PUNCT
asir-36063	31	1	first	first	ADV
asir-36063	31	2	,	,	PUNCT
asir-36063	31	3	we	we	PRON
asir-36063	31	4	define	define	VERB
asir-36063	31	5	the	the	DET
asir-36063	31	6	dsbm	dsbm	NOUN
asir-36063	31	7	as	as	SCONJ
asir-36063	31	8	follows	follow	VERB
asir-36063	31	9	.	.	PUNCT
asir-36063	32	1	given	give	VERB
asir-36063	32	2	an	an	DET
asir-36063	32	3	even	even	ADV
asir-36063	32	4	integer	integer	NOUN
asir-36063	32	5	𝑛	𝑛	PRON
asir-36063	32	6	≥	≥	NOUN
asir-36063	32	7	2	2	NUM
asir-36063	32	8	,	,	PUNCT
asir-36063	32	9	and	and	CCONJ
asir-36063	32	10	1	1	NUM
asir-36063	32	11	≥	≥	NOUN
asir-36063	32	12	𝑝	𝑝	PROPN
asir-36063	32	13	>	>	X
asir-36063	32	14	𝑞	𝑞	X
asir-36063	32	15	≥	≥	PROPN
asir-36063	32	16	0	0	NUM
asir-36063	32	17	,	,	PUNCT
asir-36063	32	18	we	we	PRON
asir-36063	32	19	say	say	VERB
asir-36063	32	20	that	that	SCONJ
asir-36063	32	21	a	a	DET
asir-36063	32	22	directed	direct	VERB
asir-36063	32	23	random	random	ADJ
asir-36063	32	24	graph	graph	NOUN
asir-36063	32	25	𝐺	𝐺	NOUN
asir-36063	32	26	is	be	AUX
asir-36063	32	27	drawn	draw	VERB
asir-36063	32	28	from	from	ADP
asir-36063	32	29	the	the	DET
asir-36063	32	30	directed	direct	VERB
asir-36063	32	31	stochastic	stochastic	ADJ
asir-36063	32	32	block	block	NOUN
asir-36063	32	33	model	model	NOUN
asir-36063	32	34	with	with	ADP
asir-36063	32	35	two	two	NUM
asir-36063	32	36	communities	community	NOUN
asir-36063	32	37	(	(	PUNCT
asir-36063	32	38	denoted	denote	VERB
asir-36063	32	39	as	as	ADP
asir-36063	32	40	dsbm⁡(𝑛	dsbm⁡(𝑛	PROPN
asir-36063	32	41	,	,	PUNCT
asir-36063	32	42	𝑝	𝑝	PROPN
asir-36063	32	43	,	,	PUNCT
asir-36063	32	44	𝑞	𝑞	NOUN
asir-36063	32	45	)	)	PUNCT
asir-36063	32	46	)	)	PUNCT
asir-36063	32	47	with	with	ADP
asir-36063	32	48	ground	ground	NOUN
asir-36063	32	49	-	-	PUNCT
asir-36063	32	50	truth	truth	NOUN
asir-36063	32	51	𝒈	𝒈	NOUN
asir-36063	32	52	,	,	PUNCT
asir-36063	32	53	if	if	SCONJ
asir-36063	32	54	𝐺	𝐺	PROPN
asir-36063	32	55	has	have	VERB
asir-36063	32	56	𝑛	𝑛	DET
asir-36063	32	57	nodes	node	NOUN
asir-36063	32	58	,	,	PUNCT
asir-36063	32	59	divided	divide	VERB
asir-36063	32	60	into	into	ADP
asir-36063	32	61	two	two	NUM
asir-36063	32	62	clusters	cluster	NOUN
asir-36063	32	63	of	of	ADP
asir-36063	32	64	𝑛/2	𝑛/2	PRON
asir-36063	32	65	nodes	nod	VERB
asir-36063	32	66	each	each	PRON
asir-36063	32	67	,	,	PUNCT
asir-36063	32	68	and	and	CCONJ
asir-36063	32	69	for	for	ADP
asir-36063	32	70	each	each	DET
asir-36063	32	71	pair	pair	NOUN
asir-36063	32	72	of	of	ADP
asir-36063	32	73	vertices	vertex	NOUN
asir-36063	32	74	(	(	PUNCT
asir-36063	32	75	𝑖	𝑖	SYM
asir-36063	32	76	,	,	PUNCT
asir-36063	32	77	𝑗	𝑗	NOUN
asir-36063	32	78	)	)	PUNCT
asir-36063	32	79	,	,	PUNCT
asir-36063	32	80	(	(	PUNCT
asir-36063	32	81	𝑖	𝑖	SYM
asir-36063	32	82	,	,	PUNCT
asir-36063	32	83	𝑗	𝑗	NOUN
asir-36063	32	84	)	)	PUNCT
asir-36063	32	85	is	be	AUX
asir-36063	32	86	an	an	DET
asir-36063	32	87	edge	edge	NOUN
asir-36063	32	88	of	of	ADP
asir-36063	32	89	𝐺	𝐺	PROPN
asir-36063	32	90	with	with	ADP
asir-36063	32	91	probability	probability	NOUN
asir-36063	32	92	𝑝	𝑝	NOUN
asir-36063	32	93	if	if	SCONJ
asir-36063	32	94	𝑖	𝑖	PROPN
asir-36063	32	95	and	and	CCONJ
asir-36063	32	96	𝑗	𝑗	INTJ
asir-36063	32	97	are	be	AUX
asir-36063	32	98	in	in	ADP
asir-36063	32	99	the	the	DET
asir-36063	32	100	same	same	ADJ
asir-36063	32	101	cluster	cluster	NOUN
asir-36063	32	102	and	and	CCONJ
asir-36063	32	103	with	with	ADP
asir-36063	32	104	probability	probability	NOUN
asir-36063	32	105	𝑞	𝑞	X
asir-36063	32	106	otherwise	otherwise	ADV
asir-36063	32	107	.	.	PUNCT
asir-36063	33	1	the	the	DET
asir-36063	33	2	𝑖-th	𝑖-th	PROPN
asir-36063	33	3	entry	entry	NOUN
asir-36063	33	4	of	of	ADP
asir-36063	33	5	𝒈	𝒈	PROPN
asir-36063	33	6	is	be	AUX
asir-36063	33	7	±1	±1	VERB
asir-36063	33	8	indicating	indicate	VERB
asir-36063	33	9	the	the	DET
asir-36063	33	10	cluster	cluster	NOUN
asir-36063	33	11	to	to	PART
asir-36063	33	12	which	which	PRON
asir-36063	33	13	the	the	DET
asir-36063	33	14	𝑖-th	𝑖-th	PROPN
asir-36063	33	15	node	node	NOUN
asir-36063	33	16	belongs	belong	VERB
asir-36063	33	17	.	.	PUNCT
asir-36063	34	1	in	in	ADP
asir-36063	34	2	particular	particular	ADJ
asir-36063	34	3	,	,	PUNCT
asir-36063	34	4	let	let	VERB
asir-36063	34	5	𝑨	𝑨	PRON
asir-36063	34	6	be	be	AUX
asir-36063	34	7	the	the	DET
asir-36063	34	8	adjacency	adjacency	NOUN
asir-36063	34	9	matrix	matrix	NOUN
asir-36063	34	10	of	of	ADP
asir-36063	34	11	𝐺.	𝐺.	NOUN
asir-36063	34	12	each	each	DET
asir-36063	34	13	entry	entry	NOUN
asir-36063	34	14	of	of	ADP
asir-36063	34	15	𝑨	𝑨	PROPN
asir-36063	34	16	is	be	AUX
asir-36063	34	17	given	give	VERB
asir-36063	34	18	by	by	ADP
asir-36063	34	19	ℙ(𝑨𝑖𝑗	ℙ(𝑨𝑖𝑗	PROPN
asir-36063	35	1	=	=	SYM
asir-36063	35	2	1	1	X
asir-36063	35	3	)	)	PUNCT
asir-36063	35	4	=	=	PRON
asir-36063	35	5	{	{	PUNCT
asir-36063	35	6	𝑝	𝑝	NOUN
asir-36063	35	7	if	if	SCONJ
asir-36063	35	8	𝑖	𝑖	PROPN
asir-36063	35	9	and	and	CCONJ
asir-36063	35	10	𝑗	𝑗	INTJ
asir-36063	35	11	are	be	AUX
asir-36063	35	12	in	in	ADP
asir-36063	35	13	the	the	DET
asir-36063	35	14	same	same	ADJ
asir-36063	35	15	community	community	NOUN
asir-36063	35	16	𝑞	𝑞	X
asir-36063	35	17	otherwise	otherwise	ADV
asir-36063	35	18	⁡1	⁡1	CCONJ
asir-36063	35	19	≤	≤	NUM
asir-36063	35	20	𝑖	𝑖	SYM
asir-36063	35	21	≤	≤	NUM
asir-36063	35	22	𝑛	𝑛	NOUN
asir-36063	35	23	,	,	PUNCT
asir-36063	35	24	1	1	NUM
asir-36063	35	25	≤	≤	NUM
asir-36063	35	26	𝑗	𝑗	PRON
asir-36063	35	27	≤	≤	ADJ
asir-36063	35	28	𝑛.	𝑛.	NOUN
asir-36063	35	29	then	then	ADV
asir-36063	35	30	the	the	DET
asir-36063	35	31	expected	expect	VERB
asir-36063	35	32	adjacency	adjacency	NOUN
asir-36063	35	33	matrix	matrix	NOUN
asir-36063	35	34	𝑨∗	𝑨∗	PROPN
asir-36063	35	35	=	=	SYM
asir-36063	35	36	𝔼𝑨	𝔼𝑨	PROPN
asir-36063	35	37	is	be	AUX
asir-36063	35	38	given	give	VERB
asir-36063	35	39	by	by	ADP
asir-36063	35	40	𝑨∗	𝑨∗	PROPN
asir-36063	35	41	=	=	SYM
asir-36063	35	42	𝔼𝑨	𝔼𝑨	PROPN
asir-36063	35	43	=	=	SYM
asir-36063	35	44	[	[	PUNCT
asir-36063	35	45	𝑝𝑱𝑛/2×𝑛/2	𝑝𝑱𝑛/2×𝑛/2	NOUN
asir-36063	35	46	𝑞𝑱𝑛/2×𝑛/2	𝑞𝑱𝑛/2×𝑛/2	PROPN
asir-36063	35	47	𝑞𝑱𝑛/2×𝑛/2	𝑞𝑱𝑛/2×𝑛/2	PROPN
asir-36063	35	48	𝑝𝑱𝑛/2×𝑛/2	𝑝𝑱𝑛/2×𝑛/2	PROPN
asir-36063	35	49	]	]	PUNCT
asir-36063	36	1	this	this	DET
asir-36063	36	2	model	model	NOUN
asir-36063	36	3	can	can	AUX
asir-36063	36	4	be	be	AUX
asir-36063	36	5	seen	see	VERB
asir-36063	36	6	as	as	ADP
asir-36063	36	7	the	the	DET
asir-36063	36	8	concatenation	concatenation	NOUN
asir-36063	36	9	of	of	ADP
asir-36063	36	10	two	two	NUM
asir-36063	36	11	directed	direct	VERB
asir-36063	36	12	erdős	erdős	NOUN
asir-36063	36	13	-	-	PUNCT
asir-36063	36	14	rényi	rényi	NOUN
asir-36063	36	15	random	random	ADJ
asir-36063	36	16	graphs	graph	NOUN
asir-36063	36	17	with	with	ADP
asir-36063	36	18	parameter	parameter	NOUN
asir-36063	36	19	𝑝	𝑝	PROPN
asir-36063	36	20	(	(	PUNCT
asir-36063	36	21	as	as	ADP
asir-36063	36	22	two	two	NUM
asir-36063	36	23	clusters	cluster	NOUN
asir-36063	36	24	)	)	PUNCT
asir-36063	36	25	and	and	CCONJ
asir-36063	36	26	the	the	DET
asir-36063	36	27	connection	connection	NOUN
asir-36063	36	28	probability	probability	NOUN
asir-36063	36	29	between	between	ADP
asir-36063	36	30	these	these	DET
asir-36063	36	31	two	two	NUM
asir-36063	36	32	graphs	graph	NOUN
asir-36063	36	33	is	be	AUX
asir-36063	36	34	𝑞.	𝑞.	ADJ
asir-36063	36	35	to	to	PART
asir-36063	36	36	theoretically	theoretically	ADV
asir-36063	36	37	understand	understand	VERB
asir-36063	36	38	community	community	NOUN
asir-36063	36	39	detection	detection	NOUN
asir-36063	36	40	in	in	ADP
asir-36063	36	41	directed	direct	VERB
asir-36063	36	42	networks	network	NOUN
asir-36063	36	43	,	,	PUNCT
asir-36063	36	44	we	we	PRON
asir-36063	36	45	are	be	AUX
asir-36063	36	46	interested	interested	ADJ
asir-36063	36	47	in	in	ADP
asir-36063	36	48	the	the	DET
asir-36063	36	49	information	information	NOUN
asir-36063	36	50	-	-	PUNCT
asir-36063	36	51	theoretical	theoretical	ADJ
asir-36063	36	52	threshold	threshold	NOUN
asir-36063	36	53	for	for	ADP
asir-36063	36	54	exact	exact	ADJ
asir-36063	36	55	recovery	recovery	NOUN
asir-36063	36	56	in	in	ADP
asir-36063	36	57	dsbm	dsbm	ADJ
asir-36063	36	58	,	,	PUNCT
asir-36063	36	59	that	that	ADV
asir-36063	36	60	is	is	ADV
asir-36063	36	61	,	,	PUNCT
asir-36063	36	62	we	we	PRON
asir-36063	36	63	want	want	VERB
asir-36063	36	64	to	to	PART
asir-36063	36	65	find	find	VERB
asir-36063	36	66	a	a	DET
asir-36063	36	67	threshold	threshold	NOUN
asir-36063	36	68	as	as	ADP
asir-36063	36	69	a	a	DET
asir-36063	36	70	function	function	NOUN
asir-36063	36	71	of	of	ADP
asir-36063	36	72	(	(	PUNCT
asir-36063	36	73	𝑛	𝑛	PROPN
asir-36063	36	74	,	,	PUNCT
asir-36063	36	75	𝑝	𝑝	NOUN
asir-36063	36	76	,	,	PUNCT
asir-36063	36	77	𝑞	𝑞	NOUN
asir-36063	36	78	)	)	PUNCT
asir-36063	36	79	,	,	PUNCT
asir-36063	36	80	above	above	ADP
asir-36063	36	81	which	which	PRON
asir-36063	36	82	exactly	exactly	ADV
asir-36063	36	83	recovering	recover	VERB
asir-36063	36	84	the	the	DET
asir-36063	36	85	membership	membership	NOUN
asir-36063	36	86	of	of	ADP
asir-36063	36	87	each	each	DET
asir-36063	36	88	node	node	NOUN
asir-36063	36	89	is	be	AUX
asir-36063	36	90	possible	possible	ADJ
asir-36063	36	91	with	with	ADP
asir-36063	36	92	probability	probability	NOUN
asir-36063	36	93	1	1	NUM
asir-36063	36	94	−	−	NOUN
asir-36063	36	95	𝑜(1	𝑜(1	NOUN
asir-36063	36	96	)	)	PUNCT
asir-36063	36	97	,	,	PUNCT
asir-36063	36	98	while	while	SCONJ
asir-36063	36	99	impossible	impossible	ADJ
asir-36063	36	100	otherwise	otherwise	ADV
asir-36063	36	101	.	.	PUNCT
asir-36063	37	1	the	the	DET
asir-36063	37	2	definition	definition	NOUN
asir-36063	37	3	of	of	ADP
asir-36063	37	4	exact	exact	ADJ
asir-36063	37	5	recovery	recovery	NOUN
asir-36063	37	6	is	be	AUX
asir-36063	37	7	stated	state	VERB
asir-36063	37	8	as	as	SCONJ
asir-36063	37	9	follows	follow	VERB
asir-36063	37	10	.	.	PUNCT
asir-36063	38	1	let	let	VERB
asir-36063	38	2	algo	algo	PROPN
asir-36063	38	3	(	(	PUNCT
asir-36063	38	4	⋅	⋅	PROPN
asir-36063	38	5	)	)	PUNCT
asir-36063	38	6	be	be	AUX
asir-36063	38	7	some	some	DET
asir-36063	38	8	community	community	NOUN
asir-36063	38	9	detection	detection	NOUN
asir-36063	38	10	algorithm	algorithm	NOUN
asir-36063	38	11	,	,	PUNCT
asir-36063	38	12	and	and	CCONJ
asir-36063	38	13	𝑨	𝑨	PROPN
asir-36063	38	14	be	be	VERB
asir-36063	38	15	the	the	DET
asir-36063	38	16	adjacency	adjacency	NOUN
asir-36063	38	17	matrix	matrix	NOUN
asir-36063	38	18	of	of	ADP
asir-36063	38	19	𝐺	𝐺	PROPN
asir-36063	38	20	∼	∼	NOUN
asir-36063	38	21	dsbm⁡(𝑛	dsbm⁡(𝑛	PROPN
asir-36063	38	22	,	,	PUNCT
asir-36063	38	23	𝑝	𝑝	PROPN
asir-36063	38	24	,	,	PUNCT
asir-36063	38	25	𝑞	𝑞	NOUN
asir-36063	38	26	)	)	PUNCT
asir-36063	38	27	with	with	ADP
asir-36063	38	28	ground	ground	NOUN
asir-36063	38	29	-	-	PUNCT
asir-36063	38	30	truth	truth	NOUN
asir-36063	38	31	membership	membership	NOUN
asir-36063	38	32	𝒈.	𝒈.	NOUN
asir-36063	38	33	then	then	ADV
asir-36063	38	34	we	we	PRON
asir-36063	38	35	say	say	VERB
asir-36063	38	36	algo⁡(⋅	algo⁡(⋅	NOUN
asir-36063	38	37	)	)	PUNCT
asir-36063	38	38	exactly	exactly	ADV
asir-36063	38	39	recovers	recover	VERB
asir-36063	38	40	the	the	DET
asir-36063	38	41	membership	membership	NOUN
asir-36063	38	42	if	if	SCONJ
asir-36063	38	43	algo⁡(𝑨	algo⁡(𝑨	PUNCT
asir-36063	38	44	)	)	PUNCT
asir-36063	38	45	=	=	SYM
asir-36063	39	1	𝒙	𝒙	X
asir-36063	39	2	=	=	SYM
asir-36063	39	3	𝒈	𝒈	X
asir-36063	39	4	where	where	SCONJ
asir-36063	39	5	𝒙	𝒙	PROPN
asir-36063	39	6	is	be	AUX
asir-36063	39	7	the	the	DET
asir-36063	39	8	membership	membership	NOUN
asir-36063	39	9	estimated	estimate	VERB
asir-36063	39	10	by	by	ADP
asir-36063	39	11	algo⁡(⋅	algo⁡(⋅	NOUN
asir-36063	39	12	)	)	PUNCT
asir-36063	39	13	.	.	PUNCT
asir-36063	40	1	since	since	SCONJ
asir-36063	40	2	it	it	PRON
asir-36063	40	3	can	can	AUX
asir-36063	40	4	be	be	AUX
asir-36063	40	5	verified	verify	VERB
asir-36063	40	6	that	that	SCONJ
asir-36063	40	7	connectedness	connectedness	NOUN
asir-36063	40	8	is	be	AUX
asir-36063	40	9	a	a	DET
asir-36063	40	10	sufficient	sufficient	ADJ
asir-36063	40	11	condition	condition	NOUN
asir-36063	40	12	for	for	ADP
asir-36063	40	13	exact	exact	ADJ
asir-36063	40	14	recovery	recovery	NOUN
asir-36063	40	15	in	in	ADP
asir-36063	40	16	dsbm	dsbm	ADJ
asir-36063	40	17	,	,	PUNCT
asir-36063	40	18	we	we	PRON
asir-36063	40	19	will	will	AUX
asir-36063	40	20	choose	choose	VERB
asir-36063	40	21	the	the	DET
asir-36063	40	22	regime	regime	NOUN
asir-36063	40	23	𝑝	𝑝	PROPN
asir-36063	40	24	=	=	SYM
asir-36063	40	25	𝑎log⁡(𝑛)/𝑛	𝑎log⁡(𝑛)/𝑛	PROPN
asir-36063	40	26	,	,	PUNCT
asir-36063	40	27	𝑞	𝑞	X
asir-36063	40	28	=	=	SYM
asir-36063	40	29	𝑏log⁡(𝑛)/𝑛	𝑏log⁡(𝑛)/𝑛	PROPN
asir-36063	40	30	in	in	ADP
asir-36063	40	31	the	the	DET
asir-36063	40	32	whole	whole	ADJ
asir-36063	40	33	paper	paper	NOUN
asir-36063	40	34	.	.	PUNCT
asir-36063	41	1	in	in	ADP
asir-36063	41	2	this	this	DET
asir-36063	41	3	work	work	NOUN
asir-36063	41	4	,	,	PUNCT
asir-36063	41	5	we	we	PRON
asir-36063	41	6	will	will	AUX
asir-36063	41	7	propose	propose	VERB
asir-36063	41	8	the	the	DET
asir-36063	41	9	threshold	threshold	NOUN
asir-36063	41	10	of	of	ADP
asir-36063	41	11	exact	exact	ADJ
asir-36063	41	12	recovery	recovery	NOUN
asir-36063	41	13	for	for	ADP
asir-36063	41	14	dsbm⁡(𝑛	dsbm⁡(𝑛	PROPN
asir-36063	41	15	,	,	PUNCT
asir-36063	41	16	𝑎	𝑎	NOUN
asir-36063	41	17	,	,	PUNCT
asir-36063	41	18	𝑏	𝑏	NOUN
asir-36063	41	19	)	)	PUNCT
asir-36063	41	20	and	and	CCONJ
asir-36063	41	21	a	a	DET
asir-36063	41	22	clustering	clustering	ADJ
asir-36063	41	23	algorithm	algorithm	NOUN
asir-36063	41	24	that	that	PRON
asir-36063	41	25	can	can	AUX
asir-36063	41	26	exactly	exactly	ADV
asir-36063	41	27	recover	recover	VERB
asir-36063	41	28	the	the	DET
asir-36063	41	29	membership	membership	NOUN
asir-36063	41	30	,	,	PUNCT
asir-36063	41	31	and	and	CCONJ
asir-36063	41	32	then	then	ADV
asir-36063	41	33	prove	prove	VERB
asir-36063	41	34	the	the	DET
asir-36063	41	35	tightness	tightness	NOUN
asir-36063	41	36	of	of	ADP
asir-36063	41	37	the	the	DET
asir-36063	41	38	threshold	threshold	NOUN
asir-36063	41	39	.	.	PUNCT
asir-36063	42	1	2.2	2.2	NUM
asir-36063	42	2	co	co	ADJ
asir-36063	42	3	-	-	ADJ
asir-36063	42	4	clustering	clustering	ADJ
asir-36063	42	5	:	:	PUNCT
asir-36063	42	6	community	community	NOUN
asir-36063	42	7	detection	detection	NOUN
asir-36063	42	8	in	in	ADP
asir-36063	42	9	directed	direct	VERB
asir-36063	42	10	networks	network	NOUN
asir-36063	42	11	co	co	VERB
asir-36063	42	12	-	-	ADJ
asir-36063	42	13	clustering	clustering	ADJ
asir-36063	42	14	was	be	AUX
asir-36063	42	15	a	a	DET
asir-36063	42	16	concept	concept	NOUN
asir-36063	42	17	first	first	ADV
asir-36063	42	18	proposed	propose	VERB
asir-36063	42	19	in	in	ADP
asir-36063	42	20	1972	1972	NUM
asir-36063	42	21	,	,	PUNCT
asir-36063	42	22	where	where	SCONJ
asir-36063	42	23	it	it	PRON
asir-36063	42	24	clusters	cluster	VERB
asir-36063	42	25	entries	entry	NOUN
asir-36063	42	26	of	of	ADP
asir-36063	42	27	a	a	DET
asir-36063	42	28	matrix	matrix	NOUN
asir-36063	42	29	∈	∈	NOUN
asir-36063	42	30	ℝ𝑛×𝑑.	ℝ𝑛×𝑑.	NOUN
asir-36063	42	31	in	in	ADP
asir-36063	42	32	the	the	DET
asir-36063	42	33	past	past	NOUN
asir-36063	42	34	,	,	PUNCT
asir-36063	42	35	co	co	ADJ
asir-36063	42	36	-	-	ADJ
asir-36063	42	37	clustering	clustering	ADJ
asir-36063	42	38	has	have	AUX
asir-36063	42	39	been	be	AUX
asir-36063	42	40	applied	apply	VERB
asir-36063	42	41	to	to	ADP
asir-36063	42	42	matrices	matrix	NOUN
asir-36063	42	43	where	where	SCONJ
asir-36063	42	44	the	the	DET
asir-36063	42	45	rows	row	NOUN
asir-36063	42	46	and	and	CCONJ
asir-36063	42	47	columns	column	NOUN
asir-36063	42	48	represent	represent	VERB
asir-36063	42	49	different	different	ADJ
asir-36063	42	50	meanings	meaning	NOUN
asir-36063	42	51	,	,	PUNCT
asir-36063	42	52	and	and	CCONJ
asir-36063	42	53	it	it	PRON
asir-36063	42	54	clusters	cluster	VERB
asir-36063	42	55	rows	row	NOUN
asir-36063	42	56	of	of	ADP
asir-36063	42	57	matrix	matrix	NOUN
asir-36063	42	58	𝑀	𝑀	PROPN
asir-36063	42	59	into	into	ADP
asir-36063	42	60	𝑘𝑟	𝑘𝑟	PROPN
asir-36063	42	61	communities	community	NOUN
asir-36063	42	62	,	,	PUNCT
asir-36063	42	63	and	and	CCONJ
asir-36063	42	64	columns	column	NOUN
asir-36063	42	65	into	into	ADP
asir-36063	42	66	𝑘𝑐	𝑘𝑐	ADP
asir-36063	42	67	communities	community	NOUN
asir-36063	42	68	.	.	PUNCT
asir-36063	43	1	for	for	ADP
asir-36063	43	2	example	example	NOUN
asir-36063	43	3	,	,	PUNCT
asir-36063	43	4	in	in	ADP
asir-36063	43	5	a	a	DET
asir-36063	43	6	matrix	matrix	NOUN
asir-36063	43	7	used	use	VERB
asir-36063	43	8	for	for	ADP
asir-36063	43	9	text	text	NOUN
asir-36063	43	10	processing	processing	NOUN
asir-36063	43	11	,	,	PUNCT
asir-36063	43	12	the	the	DET
asir-36063	43	13	rows	row	NOUN
asir-36063	43	14	represent	represent	VERB
asir-36063	43	15	documents	document	NOUN
asir-36063	43	16	,	,	PUNCT
asir-36063	43	17	and	and	CCONJ
asir-36063	43	18	the	the	DET
asir-36063	43	19	columns	column	NOUN
asir-36063	43	20	represent	represent	VERB
asir-36063	43	21	words	word	NOUN
asir-36063	43	22	.	.	PUNCT
asir-36063	44	1	therefore	therefore	ADV
asir-36063	44	2	,	,	PUNCT
asir-36063	44	3	each	each	DET
asir-36063	44	4	entry	entry	NOUN
asir-36063	44	5	in	in	ADP
asir-36063	44	6	(	(	PUNCT
asir-36063	44	7	𝑖	𝑖	SYM
asir-36063	44	8	,	,	PUNCT
asir-36063	44	9	𝑗	𝑗	NOUN
asir-36063	44	10	)	)	PUNCT
asir-36063	44	11	indicates	indicate	VERB
asir-36063	44	12	how	how	SCONJ
asir-36063	44	13	many	many	ADJ
asir-36063	44	14	time	time	NOUN
asir-36063	44	15	word	word	NOUN
asir-36063	44	16	𝑗	𝑗	PROPN
asir-36063	44	17	appears	appear	VERB
asir-36063	44	18	in	in	ADP
asir-36063	44	19	document	document	NOUN
asir-36063	44	20	𝑖.	𝑖.	ADJ
asir-36063	44	21	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	44	22	applied	apply	VERB
asir-36063	44	23	science	science	NOUN
asir-36063	44	24	and	and	CCONJ
asir-36063	44	25	innovative	innovative	ADJ
asir-36063	44	26	research	research	NOUN
asir-36063	44	27	vol	vol	NOUN
asir-36063	44	28	.	.	PROPN
asir-36063	44	29	8	8	NUM
asir-36063	44	30	,	,	PUNCT
asir-36063	44	31	no	no	INTJ
asir-36063	44	32	.	.	NOUN
asir-36063	44	33	1	1	NUM
asir-36063	44	34	,	,	PUNCT
asir-36063	44	35	2024	2024	NUM
asir-36063	44	36	66	66	NUM
asir-36063	44	37	published	publish	VERB
asir-36063	44	38	by	by	ADP
asir-36063	44	39	scholink	scholink	PROPN
asir-36063	44	40	inc	inc	PROPN
asir-36063	44	41	.	.	PROPN
asir-36063	44	42	however	however	ADV
asir-36063	44	43	,	,	PUNCT
asir-36063	44	44	in	in	ADP
asir-36063	44	45	this	this	DET
asir-36063	44	46	thesis	thesis	NOUN
asir-36063	44	47	,	,	PUNCT
asir-36063	44	48	we	we	PRON
asir-36063	44	49	apply	apply	VERB
asir-36063	44	50	co	co	VERB
asir-36063	44	51	-	-	ADJ
asir-36063	44	52	clustering	clustering	ADJ
asir-36063	44	53	to	to	ADP
asir-36063	44	54	a	a	DET
asir-36063	44	55	matrix	matrix	NOUN
asir-36063	44	56	where	where	SCONJ
asir-36063	44	57	the	the	DET
asir-36063	44	58	rows	row	NOUN
asir-36063	44	59	and	and	CCONJ
asir-36063	44	60	columns	column	NOUN
asir-36063	44	61	index	index	NOUN
asir-36063	44	62	the	the	DET
asir-36063	44	63	same	same	ADJ
asir-36063	44	64	set	set	NOUN
asir-36063	44	65	of	of	ADP
asir-36063	44	66	vertex	vertex	NOUN
asir-36063	44	67	.	.	PUNCT
asir-36063	45	1	specifically	specifically	ADV
asir-36063	45	2	,	,	PUNCT
asir-36063	45	3	the	the	DET
asir-36063	45	4	𝑖th	𝑖th	NOUN
asir-36063	45	5	row	row	NOUN
asir-36063	45	6	of	of	ADP
asir-36063	45	7	the	the	DET
asir-36063	45	8	matrix	matrix	NOUN
asir-36063	45	9	represents	represent	VERB
asir-36063	45	10	the	the	DET
asir-36063	45	11	connection	connection	NOUN
asir-36063	45	12	of	of	ADP
asir-36063	45	13	the	the	DET
asir-36063	45	14	𝑖th	𝑖th	PROPN
asir-36063	45	15	vertex	vertex	NOUN
asir-36063	45	16	,	,	PUNCT
asir-36063	45	17	where	where	SCONJ
asir-36063	45	18	it	it	PRON
asir-36063	45	19	shows	show	VERB
asir-36063	45	20	the	the	DET
asir-36063	45	21	outgoing	outgoing	ADJ
asir-36063	45	22	edges	edge	NOUN
asir-36063	45	23	for	for	ADP
asir-36063	45	24	vertex	vertex	NOUN
asir-36063	45	25	𝑖.	𝑖.	VERB
asir-36063	45	26	the	the	DET
asir-36063	45	27	𝑖th	𝑖th	NOUN
asir-36063	45	28	column	column	NOUN
asir-36063	45	29	of	of	ADP
asir-36063	45	30	the	the	DET
asir-36063	45	31	matrix	matrix	NOUN
asir-36063	45	32	represents	represent	VERB
asir-36063	45	33	the	the	DET
asir-36063	45	34	incoming	incoming	ADJ
asir-36063	45	35	edges	edge	NOUN
asir-36063	45	36	of	of	ADP
asir-36063	45	37	𝑖th	𝑖th	X
asir-36063	45	38	vertex	vertex	NOUN
asir-36063	45	39	.	.	PUNCT
asir-36063	46	1	therefore	therefore	ADV
asir-36063	46	2	,	,	PUNCT
asir-36063	46	3	each	each	DET
asir-36063	46	4	vertex	vertex	NOUN
asir-36063	46	5	𝑖	𝑖	X
asir-36063	46	6	is	be	AUX
asir-36063	46	7	included	include	VERB
asir-36063	46	8	in	in	ADP
asir-36063	46	9	two	two	NUM
asir-36063	46	10	communities	community	NOUN
asir-36063	46	11	,	,	PUNCT
asir-36063	46	12	one	one	NUM
asir-36063	46	13	for	for	ADP
asir-36063	46	14	the	the	DET
asir-36063	46	15	row	row	NOUN
asir-36063	46	16	and	and	CCONJ
asir-36063	46	17	one	one	NUM
asir-36063	46	18	for	for	ADP
asir-36063	46	19	the	the	DET
asir-36063	46	20	column	column	NOUN
asir-36063	46	21	.	.	PUNCT
asir-36063	47	1	it	it	PRON
asir-36063	47	2	is	be	AUX
asir-36063	47	3	worth	worth	ADJ
asir-36063	47	4	mentioning	mention	VERB
asir-36063	47	5	that	that	SCONJ
asir-36063	47	6	due	due	ADP
asir-36063	47	7	to	to	ADP
asir-36063	47	8	the	the	DET
asir-36063	47	9	directedness	directedness	NOUN
asir-36063	47	10	of	of	ADP
asir-36063	47	11	the	the	DET
asir-36063	47	12	dsbm	dsbm	NOUN
asir-36063	47	13	,	,	PUNCT
asir-36063	47	14	the	the	DET
asir-36063	47	15	connectedness	connectedness	NOUN
asir-36063	47	16	of	of	ADP
asir-36063	47	17	the	the	DET
asir-36063	47	18	row	row	NOUN
asir-36063	47	19	community	community	NOUN
asir-36063	47	20	and	and	CCONJ
asir-36063	47	21	the	the	DET
asir-36063	47	22	column	column	NOUN
asir-36063	47	23	community	community	NOUN
asir-36063	47	24	of	of	ADP
asir-36063	47	25	𝑖th	𝑖th	X
asir-36063	47	26	vertex	vertex	NOUN
asir-36063	47	27	is	be	AUX
asir-36063	47	28	not	not	PART
asir-36063	47	29	necessarily	necessarily	ADV
asir-36063	47	30	the	the	DET
asir-36063	47	31	same	same	ADJ
asir-36063	47	32	.	.	PUNCT
asir-36063	48	1	specifically	specifically	ADV
asir-36063	48	2	,	,	PUNCT
asir-36063	48	3	we	we	PRON
asir-36063	48	4	would	would	AUX
asir-36063	48	5	like	like	VERB
asir-36063	48	6	to	to	PART
asir-36063	48	7	find	find	VERB
asir-36063	48	8	the	the	DET
asir-36063	48	9	maximum	maximum	ADJ
asir-36063	48	10	likelihood	likelihood	NOUN
asir-36063	48	11	estimation	estimation	NOUN
asir-36063	48	12	(	(	PUNCT
asir-36063	48	13	mle	mle	PROPN
asir-36063	48	14	)	)	PUNCT
asir-36063	48	15	to	to	ADP
asir-36063	48	16	the	the	DET
asir-36063	48	17	communities	community	NOUN
asir-36063	48	18	in	in	ADP
asir-36063	48	19	the	the	DET
asir-36063	48	20	dsbm	dsbm	NOUN
asir-36063	48	21	.	.	PUNCT
asir-36063	49	1	we	we	PRON
asir-36063	49	2	assume	assume	VERB
asir-36063	49	3	𝑢	𝑢	PRON
asir-36063	49	4	and	and	CCONJ
asir-36063	49	5	𝑣	𝑣	PRON
asir-36063	49	6	to	to	PART
asir-36063	49	7	be	be	AUX
asir-36063	49	8	n	n	PRON
asir-36063	49	9	by	by	ADP
asir-36063	49	10	1	1	NUM
asir-36063	49	11	matrix	matrix	NOUN
asir-36063	49	12	,	,	PUNCT
asir-36063	49	13	where	where	SCONJ
asir-36063	49	14	the	the	DET
asir-36063	49	15	first	first	ADJ
asir-36063	49	16	𝑛	𝑛	DET
asir-36063	49	17	2	2	NUM
asir-36063	49	18	entries	entry	NOUN
asir-36063	49	19	are	be	AUX
asir-36063	49	20	1	1	NUM
asir-36063	49	21	,	,	PUNCT
asir-36063	49	22	and	and	CCONJ
asir-36063	49	23	others	other	NOUN
asir-36063	49	24	are	be	AUX
asir-36063	49	25	-1	-1	PUNCT
asir-36063	49	26	.	.	PUNCT
asir-36063	50	1	we	we	PRON
asir-36063	50	2	have	have	VERB
asir-36063	50	3	𝑨	𝑨	NOUN
asir-36063	50	4	as	as	ADP
asir-36063	50	5	our	our	PRON
asir-36063	50	6	adjacency	adjacency	NOUN
asir-36063	50	7	matrix	matrix	NOUN
asir-36063	50	8	,	,	PUNCT
asir-36063	50	9	which	which	PRON
asir-36063	50	10	reflects	reflect	VERB
asir-36063	50	11	the	the	DET
asir-36063	50	12	realistic	realistic	ADJ
asir-36063	50	13	connection	connection	NOUN
asir-36063	50	14	behavior	behavior	NOUN
asir-36063	50	15	.	.	PUNCT
asir-36063	51	1	then	then	ADV
asir-36063	51	2	we	we	PRON
asir-36063	51	3	have	have	VERB
asir-36063	51	4	the	the	DET
asir-36063	51	5	following	follow	VERB
asir-36063	51	6	equation	equation	NOUN
asir-36063	51	7	:	:	PUNCT
asir-36063	51	8	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max⁡⁡⁡⁡𝒖⊤𝑨𝒗	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max⁡⁡⁡⁡𝒖⊤𝑨𝒗	ADP
asir-36063	51	9	s.t	s.t	PROPN
asir-36063	51	10	⁡⁡⁡⁡𝑢𝑖	⁡⁡⁡⁡𝑢𝑖	PROPN
asir-36063	51	11	=	=	PUNCT
asir-36063	51	12	±1,1	±1,1	CCONJ
asir-36063	51	13	≤	≤	NUM
asir-36063	51	14	𝑖	𝑖	SYM
asir-36063	51	15	≤	≤	NUM
asir-36063	51	16	𝑛,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.1	𝑛,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.1	NUM
asir-36063	51	17	）	）	NOUN
asir-36063	51	18	⁡⁡⁡⁡𝑣𝑖	⁡⁡⁡⁡𝑣𝑖	NOUN
asir-36063	51	19	=	=	PUNCT
asir-36063	51	20	±1,1	±1,1	CCONJ
asir-36063	51	21	≤	≤	NUM
asir-36063	51	22	𝑖	𝑖	SYM
asir-36063	51	23	≤	≤	NOUN
asir-36063	51	24	𝑛.	𝑛.	NOUN
asir-36063	51	25	in	in	ADP
asir-36063	51	26	this	this	DET
asir-36063	51	27	multiplication	multiplication	NOUN
asir-36063	51	28	,	,	PUNCT
asir-36063	51	29	vertices	vertice	VERB
asir-36063	51	30	within	within	ADP
asir-36063	51	31	the	the	DET
asir-36063	51	32	same	same	ADJ
asir-36063	51	33	community	community	NOUN
asir-36063	51	34	would	would	AUX
asir-36063	51	35	give	give	VERB
asir-36063	51	36	a	a	DET
asir-36063	51	37	positive	positive	ADJ
asir-36063	51	38	value	value	NOUN
asir-36063	51	39	,	,	PUNCT
asir-36063	51	40	and	and	CCONJ
asir-36063	51	41	we	we	PRON
asir-36063	51	42	try	try	VERB
asir-36063	51	43	to	to	PART
asir-36063	51	44	maximize	maximize	VERB
asir-36063	51	45	the	the	DET
asir-36063	51	46	value	value	NOUN
asir-36063	51	47	for	for	ADP
asir-36063	51	48	all	all	DET
asir-36063	51	49	vertices	vertex	NOUN
asir-36063	51	50	.	.	PUNCT
asir-36063	52	1	however	however	ADV
asir-36063	52	2	,	,	PUNCT
asir-36063	52	3	this	this	PRON
asir-36063	52	4	still	still	ADV
asir-36063	52	5	remains	remain	VERB
asir-36063	52	6	a	a	DET
asir-36063	52	7	very	very	ADV
asir-36063	52	8	challenging	challenging	ADJ
asir-36063	52	9	problem	problem	NOUN
asir-36063	52	10	to	to	PART
asir-36063	52	11	tackle	tackle	VERB
asir-36063	52	12	as	as	SCONJ
asir-36063	52	13	the	the	DET
asir-36063	52	14	conditions	condition	NOUN
asir-36063	52	15	remain	remain	VERB
asir-36063	52	16	discrete	discrete	ADJ
asir-36063	52	17	.	.	PUNCT
asir-36063	53	1	therefore	therefore	ADV
asir-36063	53	2	,	,	PUNCT
asir-36063	53	3	we	we	PRON
asir-36063	53	4	need	need	VERB
asir-36063	53	5	to	to	PART
asir-36063	53	6	use	use	VERB
asir-36063	53	7	semi	semi	ADJ
asir-36063	53	8	-	-	ADJ
asir-36063	53	9	definite	definite	ADJ
asir-36063	53	10	programming	programming	NOUN
asir-36063	53	11	(	(	PUNCT
asir-36063	53	12	sdp	sdp	NOUN
asir-36063	53	13	)	)	PUNCT
asir-36063	53	14	algorithm	algorithm	NOUN
asir-36063	53	15	to	to	PART
asir-36063	53	16	loosen	loosen	VERB
asir-36063	53	17	the	the	DET
asir-36063	53	18	conditions	condition	NOUN
asir-36063	53	19	and	and	CCONJ
asir-36063	53	20	find	find	VERB
asir-36063	53	21	the	the	DET
asir-36063	53	22	solution	solution	NOUN
asir-36063	53	23	under	under	ADP
asir-36063	53	24	that	that	DET
asir-36063	53	25	condition	condition	NOUN
asir-36063	53	26	,	,	PUNCT
asir-36063	53	27	and	and	CCONJ
asir-36063	53	28	lastly	lastly	ADV
asir-36063	53	29	check	check	VERB
asir-36063	53	30	whether	whether	SCONJ
asir-36063	53	31	the	the	DET
asir-36063	53	32	solution	solution	NOUN
asir-36063	53	33	would	would	AUX
asir-36063	53	34	work	work	VERB
asir-36063	53	35	under	under	ADP
asir-36063	53	36	the	the	DET
asir-36063	53	37	initial	initial	ADJ
asir-36063	53	38	condition	condition	NOUN
asir-36063	53	39	.	.	PUNCT
asir-36063	54	1	the	the	DET
asir-36063	54	2	detailed	detailed	ADJ
asir-36063	54	3	process	process	NOUN
asir-36063	54	4	will	will	AUX
asir-36063	54	5	be	be	AUX
asir-36063	54	6	further	far	ADV
asir-36063	54	7	explained	explain	VERB
asir-36063	54	8	in	in	ADP
asir-36063	54	9	the	the	DET
asir-36063	54	10	following	follow	VERB
asir-36063	54	11	section	section	NOUN
asir-36063	54	12	.	.	PUNCT
asir-36063	55	1	2.3	2.3	NUM
asir-36063	55	2	sdp	sdp	NOUN
asir-36063	55	3	relaxation	relaxation	NOUN
asir-36063	55	4	for	for	ADP
asir-36063	55	5	co	co	NOUN
asir-36063	55	6	-	-	ADJ
asir-36063	55	7	clustering	cluster	VERB
asir-36063	55	8	the	the	DET
asir-36063	55	9	programming	programming	NOUN
asir-36063	55	10	(	(	PUNCT
asir-36063	55	11	2.1	2.1	NUM
asir-36063	55	12	)	)	PUNCT
asir-36063	55	13	is	be	AUX
asir-36063	55	14	indeed	indeed	ADV
asir-36063	55	15	finding	find	VERB
asir-36063	55	16	the	the	DET
asir-36063	55	17	maximum	maximum	ADJ
asir-36063	55	18	likelihood	likelihood	NOUN
asir-36063	55	19	estimation	estimation	NOUN
asir-36063	55	20	to	to	ADP
asir-36063	55	21	the	the	DET
asir-36063	55	22	membership	membership	NOUN
asir-36063	55	23	of	of	ADP
asir-36063	55	24	the	the	DET
asir-36063	55	25	nodes	node	NOUN
asir-36063	55	26	,	,	PUNCT
asir-36063	55	27	but	but	CCONJ
asir-36063	55	28	it	it	PRON
asir-36063	55	29	is	be	AUX
asir-36063	55	30	challenging	challenge	VERB
asir-36063	55	31	due	due	ADJ
asir-36063	55	32	to	to	ADP
asir-36063	55	33	np	np	INTJ
asir-36063	55	34	-	-	PUNCT
asir-36063	55	35	hardness	hardness	NOUN
asir-36063	55	36	.	.	PUNCT
asir-36063	56	1	we	we	PRON
asir-36063	56	2	can	can	AUX
asir-36063	56	3	simplify	simplify	VERB
asir-36063	56	4	the	the	DET
asir-36063	56	5	algorithm	algorithm	NOUN
asir-36063	56	6	(	(	PUNCT
asir-36063	56	7	2.1	2.1	NUM
asir-36063	56	8	)	)	PUNCT
asir-36063	56	9	into	into	ADP
asir-36063	56	10	:	:	PUNCT
asir-36063	56	11	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max	PROPN
asir-36063	56	12	tr⁡(𝒖⊤𝑨𝒗	tr⁡(𝒖⊤𝑨𝒗	PROPN
asir-36063	56	13	)	)	PUNCT
asir-36063	56	14	s.t	s.t	PROPN
asir-36063	56	15	𝑢𝑖	𝑢𝑖	X
asir-36063	56	16	=	=	X
asir-36063	56	17	±1,1	±1,1	CCONJ
asir-36063	56	18	≤	≤	NUM
asir-36063	56	19	𝑖	𝑖	SYM
asir-36063	56	20	≤	≤	NUM
asir-36063	56	21	𝑛	𝑛	NOUN
asir-36063	56	22	,	,	PUNCT
asir-36063	56	23	𝑣𝑖	𝑣𝑖	ADP
asir-36063	56	24	=	=	SYM
asir-36063	56	25	±1,1	±1,1	CCONJ
asir-36063	56	26	≤	≤	NUM
asir-36063	56	27	𝑖	𝑖	SYM
asir-36063	56	28	≤	≤	NOUN
asir-36063	56	29	𝑛.	𝑛.	NOUN
asir-36063	56	30	（	（	NOUN
asir-36063	56	31	2.2	2.2	NUM
asir-36063	56	32	）	）	PUNCT
asir-36063	56	33	however	however	ADV
asir-36063	56	34	,	,	PUNCT
asir-36063	56	35	finding	find	VERB
asir-36063	56	36	the	the	DET
asir-36063	56	37	row	row	NOUN
asir-36063	56	38	membership	membership	NOUN
asir-36063	56	39	vector	vector	NOUN
asir-36063	56	40	𝒖	𝒖	X
asir-36063	56	41	and	and	CCONJ
asir-36063	56	42	column	column	NOUN
asir-36063	56	43	membership	membership	NOUN
asir-36063	56	44	vector	vector	NOUN
asir-36063	56	45	𝒗	𝒗	PROPN
asir-36063	56	46	from	from	ADP
asir-36063	56	47	this	this	DET
asir-36063	56	48	programming	programming	NOUN
asir-36063	56	49	is	be	AUX
asir-36063	56	50	np	np	NOUN
asir-36063	56	51	-	-	PUNCT
asir-36063	56	52	hard	hard	ADJ
asir-36063	56	53	due	due	ADP
asir-36063	56	54	to	to	ADP
asir-36063	56	55	the	the	DET
asir-36063	56	56	following	following	ADJ
asir-36063	56	57	reasons	reason	NOUN
asir-36063	56	58	:	:	PUNCT
asir-36063	56	59	(	(	PUNCT
asir-36063	56	60	1	1	X
asir-36063	56	61	)	)	PUNCT
asir-36063	56	62	𝑨	𝑨	NOUN
asir-36063	56	63	is	be	AUX
asir-36063	56	64	asymmetric	asymmetric	ADJ
asir-36063	56	65	,	,	PUNCT
asir-36063	56	66	so	so	SCONJ
asir-36063	56	67	there	there	PRON
asir-36063	56	68	are	be	VERB
asir-36063	56	69	limited	limited	ADJ
asir-36063	56	70	linear	linear	ADJ
asir-36063	56	71	algebra	algebra	NOUN
asir-36063	56	72	algorithms	algorithms	NOUN
asir-36063	56	73	that	that	PRON
asir-36063	56	74	can	can	AUX
asir-36063	56	75	be	be	AUX
asir-36063	56	76	used	use	VERB
asir-36063	56	77	;	;	PUNCT
asir-36063	56	78	(	(	PUNCT
asir-36063	56	79	2	2	X
asir-36063	56	80	)	)	PUNCT
asir-36063	56	81	the	the	DET
asir-36063	56	82	problem	problem	NOUN
asir-36063	56	83	is	be	AUX
asir-36063	56	84	nonconvex	nonconvex	NOUN
asir-36063	56	85	;	;	PUNCT
asir-36063	56	86	(	(	PUNCT
asir-36063	56	87	3	3	X
asir-36063	56	88	)	)	PUNCT
asir-36063	56	89	there	there	PRON
asir-36063	56	90	are	be	VERB
asir-36063	56	91	limited	limit	VERB
asir-36063	56	92	prior	prior	ADJ
asir-36063	56	93	knowledge	knowledge	NOUN
asir-36063	56	94	about	about	ADP
asir-36063	56	95	this	this	DET
asir-36063	56	96	model	model	NOUN
asir-36063	56	97	and	and	CCONJ
asir-36063	56	98	there	there	PRON
asir-36063	56	99	are	be	VERB
asir-36063	56	100	no	no	DET
asir-36063	56	101	constraints	constraint	NOUN
asir-36063	56	102	in	in	ADP
asir-36063	56	103	the	the	DET
asir-36063	56	104	model	model	NOUN
asir-36063	56	105	in	in	ADP
asir-36063	56	106	(	(	PUNCT
asir-36063	56	107	2.2	2.2	NUM
asir-36063	56	108	)	)	PUNCT
asir-36063	56	109	.	.	PUNCT
asir-36063	57	1	in	in	ADP
asir-36063	57	2	order	order	NOUN
asir-36063	57	3	to	to	PART
asir-36063	57	4	tackle	tackle	VERB
asir-36063	57	5	the	the	DET
asir-36063	57	6	third	third	ADJ
asir-36063	57	7	reason	reason	NOUN
asir-36063	57	8	,	,	PUNCT
asir-36063	57	9	we	we	PRON
asir-36063	57	10	know	know	VERB
asir-36063	57	11	that	that	SCONJ
asir-36063	57	12	as	as	SCONJ
asir-36063	57	13	defined	define	VERB
asir-36063	57	14	,	,	PUNCT
asir-36063	57	15	𝒖	𝒖	NOUN
asir-36063	57	16	and	and	CCONJ
asir-36063	57	17	𝒗	𝒗	PROPN
asir-36063	57	18	are	be	AUX
asir-36063	57	19	perpendicular	perpendicular	ADJ
asir-36063	57	20	to	to	ADP
asir-36063	57	21	𝟏	𝟏	NUM
asir-36063	57	22	matrix	matrix	VERB
asir-36063	57	23	,	,	PUNCT
asir-36063	57	24	so	so	CCONJ
asir-36063	57	25	their	their	PRON
asir-36063	57	26	dot	dot	NOUN
asir-36063	57	27	product	product	NOUN
asir-36063	57	28	would	would	AUX
asir-36063	57	29	equal	equal	VERB
asir-36063	57	30	0	0	NUM
asir-36063	57	31	.	.	PUNCT
asir-36063	58	1	therefore	therefore	ADV
asir-36063	58	2	,	,	PUNCT
asir-36063	58	3	we	we	PRON
asir-36063	58	4	can	can	AUX
asir-36063	58	5	penalize	penalize	VERB
asir-36063	58	6	algorithm	algorithm	NOUN
asir-36063	58	7	(	(	PUNCT
asir-36063	58	8	2.2	2.2	NUM
asir-36063	58	9	)	)	PUNCT
asir-36063	58	10	with	with	ADP
asir-36063	58	11	𝑢𝑇𝟏𝑛	𝑢𝑇𝟏𝑛	NOUN
asir-36063	58	12	and	and	CCONJ
asir-36063	58	13	𝑣𝑇𝟏𝑛	𝑣𝑇𝟏𝑛	NOUN
asir-36063	58	14	to	to	ADP
asir-36063	58	15	the	the	DET
asir-36063	58	16	following	follow	VERB
asir-36063	58	17	function	function	NOUN
asir-36063	58	18	:	:	PUNCT
asir-36063	58	19	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max	PROPN
asir-36063	58	20	tr⁡(𝒖⊤𝑨𝒗	tr⁡(𝒖⊤𝑨𝒗	PROPN
asir-36063	58	21	)	)	PUNCT
asir-36063	58	22	−	−	PROPN
asir-36063	58	23	𝜆(⟨𝒖	𝜆(⟨𝒖	ADJ
asir-36063	58	24	,	,	PUNCT
asir-36063	58	25	𝟏𝑛/√𝑛⟩	𝟏𝑛/√𝑛⟩	PROPN
asir-36063	58	26	+	+	CCONJ
asir-36063	58	27	⟨𝒗	⟨𝒗	PROPN
asir-36063	58	28	,	,	PUNCT
asir-36063	58	29	𝟏𝑛/√𝑛⟩	𝟏𝑛/√𝑛⟩	NOUN
asir-36063	58	30	)	)	PUNCT
asir-36063	58	31	s.t	s.t	PROPN
asir-36063	58	32	𝑢𝑖	𝑢𝑖	NOUN
asir-36063	58	33	=	=	X
asir-36063	58	34	±1,1	±1,1	CCONJ
asir-36063	58	35	≤	≤	NUM
asir-36063	58	36	𝑖	𝑖	SYM
asir-36063	58	37	≤	≤	NUM
asir-36063	58	38	𝑛	𝑛	NOUN
asir-36063	58	39	,	,	PUNCT
asir-36063	58	40	𝑣𝑖	𝑣𝑖	ADP
asir-36063	58	41	=	=	SYM
asir-36063	58	42	±1,1	±1,1	CCONJ
asir-36063	58	43	≤	≤	NUM
asir-36063	58	44	𝑖	𝑖	SYM
asir-36063	58	45	≤	≤	NOUN
asir-36063	58	46	𝑛.	𝑛.	NOUN
asir-36063	58	47	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.3	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.3	PROPN
asir-36063	58	48	）	）	NOUN
asir-36063	58	49	another	another	DET
asir-36063	58	50	key	key	ADJ
asir-36063	58	51	characteristic	characteristic	NOUN
asir-36063	58	52	of	of	ADP
asir-36063	58	53	dsbm	dsbm	ADJ
asir-36063	58	54	is	be	AUX
asir-36063	58	55	its	its	PRON
asir-36063	58	56	directness	directness	NOUN
asir-36063	58	57	,	,	PUNCT
asir-36063	58	58	and	and	CCONJ
asir-36063	58	59	in	in	ADP
asir-36063	58	60	order	order	NOUN
asir-36063	58	61	to	to	PART
asir-36063	58	62	deal	deal	VERB
asir-36063	58	63	with	with	ADP
asir-36063	58	64	the	the	DET
asir-36063	58	65	directness	directness	NOUN
asir-36063	58	66	of	of	ADP
asir-36063	58	67	𝐺	𝐺	PROPN
asir-36063	58	68	,	,	PUNCT
asir-36063	58	69	we	we	PRON
asir-36063	58	70	need	need	VERB
asir-36063	58	71	to	to	PART
asir-36063	58	72	consider	consider	VERB
asir-36063	58	73	the	the	DET
asir-36063	58	74	symmetric	symmetric	ADJ
asir-36063	58	75	augmented	augment	VERB
asir-36063	58	76	matrix	matrix	NOUN
asir-36063	58	77	defined	define	VERB
asir-36063	58	78	below	below	ADV
asir-36063	58	79	to	to	PART
asir-36063	58	80	represent	represent	VERB
asir-36063	58	81	the	the	DET
asir-36063	58	82	direction	direction	NOUN
asir-36063	58	83	:	:	PUNCT
asir-36063	58	84	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	58	85	applied	apply	VERB
asir-36063	58	86	science	science	NOUN
asir-36063	58	87	and	and	CCONJ
asir-36063	58	88	innovative	innovative	ADJ
asir-36063	58	89	research	research	NOUN
asir-36063	58	90	vol	vol	NOUN
asir-36063	58	91	.	.	PROPN
asir-36063	59	1	8	8	NUM
asir-36063	59	2	,	,	PUNCT
asir-36063	59	3	no	no	INTJ
asir-36063	59	4	.	.	NOUN
asir-36063	59	5	1	1	NUM
asir-36063	59	6	,	,	PUNCT
asir-36063	59	7	2024	2024	NUM
asir-36063	59	8	67	67	NUM
asir-36063	59	9	published	publish	VERB
asir-36063	59	10	by	by	ADP
asir-36063	59	11	scholink	scholink	PROPN
asir-36063	59	12	inc	inc	PROPN
asir-36063	59	13	.	.	PROPN
asir-36063	59	14	�	�	PROPN
asir-36063	59	15	̃	̃	PROPN
asir-36063	59	16	�	�	NOUN
asir-36063	59	17	=	=	SYM
asir-36063	59	18	[	[	PUNCT
asir-36063	59	19	𝟎𝑛×𝑛	𝟎𝑛×𝑛	PUNCT
asir-36063	59	20	𝑨⊤	𝑨⊤	NUM
asir-36063	59	21	𝑨	𝑨	PROPN
asir-36063	59	22	𝟎𝑛×𝑛	𝟎𝑛×𝑛	PUNCT
asir-36063	59	23	]	]	PUNCT
asir-36063	59	24	.	.	PUNCT
asir-36063	60	1	given	give	VERB
asir-36063	60	2	the	the	DET
asir-36063	60	3	singular	singular	ADJ
asir-36063	60	4	value	value	NOUN
asir-36063	60	5	decomposition	decomposition	NOUN
asir-36063	60	6	(	(	PUNCT
asir-36063	60	7	svd	svd	PROPN
asir-36063	60	8	)	)	PUNCT
asir-36063	60	9	of	of	ADP
asir-36063	60	10	𝑨	𝑨	PROPN
asir-36063	60	11	being	be	AUX
asir-36063	60	12	𝑨	𝑨	NOUN
asir-36063	60	13	=	=	SYM
asir-36063	60	14	𝑼𝚺𝑽⊤	𝑼𝚺𝑽⊤	PROPN
asir-36063	60	15	,	,	PUNCT
asir-36063	60	16	then	then	ADV
asir-36063	60	17	the	the	DET
asir-36063	60	18	eigen	eigen	NOUN
asir-36063	60	19	-	-	PUNCT
asir-36063	60	20	decomposition	decomposition	NOUN
asir-36063	60	21	of	of	ADP
asir-36063	60	22	�	�	PROPN
asir-36063	60	23	̃	̃	PROPN
asir-36063	60	24	�	�	PROPN
asir-36063	60	25	can	can	AUX
asir-36063	60	26	be	be	AUX
asir-36063	60	27	formulated	formulate	VERB
asir-36063	60	28	as	as	ADP
asir-36063	60	29	�	�	PROPN
asir-36063	60	30	̃	̃	PROPN
asir-36063	60	31	�	�	NOUN
asir-36063	60	32	=	=	SYM
asir-36063	60	33	1	1	NUM
asir-36063	60	34	√2	√2	PROPN
asir-36063	60	35	[	[	PUNCT
asir-36063	60	36	𝑽	𝑽	PROPN
asir-36063	60	37	𝑽	𝑽	PROPN
asir-36063	60	38	𝑼	𝑼	PROPN
asir-36063	60	39	−𝑼	−𝑼	NOUN
asir-36063	60	40	]	]	PUNCT
asir-36063	60	41	[	[	PUNCT
asir-36063	60	42	𝚺	𝚺	ADV
asir-36063	60	43	𝟎𝑛×𝑛	𝟎𝑛×𝑛	PUNCT
asir-36063	60	44	𝟎𝑛×𝑛	𝟎𝑛×𝑛	PUNCT
asir-36063	61	1	−𝚺	−𝚺	PROPN
asir-36063	61	2	]	]	PUNCT
asir-36063	61	3	1	1	NUM
asir-36063	61	4	√2	√2	NOUN
asir-36063	61	5	[	[	PUNCT
asir-36063	61	6	𝑽	𝑽	PROPN
asir-36063	61	7	⊤	⊤	NOUN
asir-36063	61	8	𝑼⊤	𝑼⊤	NUM
asir-36063	61	9	𝑽⊤	𝑽⊤	VERB
asir-36063	61	10	−𝑼⊤	−𝑼⊤	NOUN
asir-36063	61	11	]	]	PUNCT
asir-36063	61	12	.	.	PUNCT
asir-36063	62	1	it	it	PRON
asir-36063	62	2	is	be	AUX
asir-36063	62	3	worth	worth	ADJ
asir-36063	62	4	noting	note	VERB
asir-36063	62	5	that	that	SCONJ
asir-36063	62	6	conditions	condition	NOUN
asir-36063	62	7	for	for	ADP
asir-36063	62	8	algorithm	algorithm	NOUN
asir-36063	62	9	(	(	PUNCT
asir-36063	62	10	2.3	2.3	NUM
asir-36063	62	11	)	)	PUNCT
asir-36063	62	12	is	be	AUX
asir-36063	62	13	discrete	discrete	ADJ
asir-36063	62	14	,	,	PUNCT
asir-36063	62	15	where	where	SCONJ
asir-36063	62	16	the	the	DET
asir-36063	62	17	algorithm	algorithm	NOUN
asir-36063	62	18	is	be	AUX
asir-36063	62	19	unsolvable	unsolvable	ADJ
asir-36063	62	20	in	in	ADP
asir-36063	62	21	polynomial	polynomial	ADJ
asir-36063	62	22	time	time	NOUN
asir-36063	62	23	.	.	PUNCT
asir-36063	63	1	hence	hence	ADV
asir-36063	63	2	,	,	PUNCT
asir-36063	63	3	we	we	PRON
asir-36063	63	4	need	need	VERB
asir-36063	63	5	to	to	PART
asir-36063	63	6	loosen	loosen	VERB
asir-36063	63	7	the	the	DET
asir-36063	63	8	constraints	constraint	NOUN
asir-36063	63	9	using	use	VERB
asir-36063	63	10	sdp	sdp	NOUN
asir-36063	63	11	,	,	PUNCT
asir-36063	63	12	converting	convert	VERB
asir-36063	63	13	them	they	PRON
asir-36063	63	14	into	into	ADP
asir-36063	63	15	semi	semi	ADJ
asir-36063	63	16	-	-	ADJ
asir-36063	63	17	definite	definite	ADJ
asir-36063	63	18	constraints	constraint	NOUN
asir-36063	63	19	that	that	PRON
asir-36063	63	20	would	would	AUX
asir-36063	63	21	be	be	AUX
asir-36063	63	22	solvable	solvable	ADJ
asir-36063	63	23	in	in	ADP
asir-36063	63	24	polynomial	polynomial	ADJ
asir-36063	63	25	time	time	NOUN
asir-36063	63	26	.	.	PUNCT
asir-36063	64	1	using	use	VERB
asir-36063	64	2	the	the	DET
asir-36063	64	3	wellknown	wellknown	ADJ
asir-36063	64	4	goemans	goemans	PROPN
asir-36063	64	5	-	-	PUNCT
asir-36063	64	6	williams	williams	PROPN
asir-36063	64	7	relaxation	relaxation	NOUN
asir-36063	64	8	,	,	PUNCT
asir-36063	64	9	we	we	PRON
asir-36063	64	10	can	can	AUX
asir-36063	64	11	formulate	formulate	VERB
asir-36063	64	12	semi	semi	ADJ
asir-36063	64	13	-	-	ADJ
asir-36063	64	14	definite	definite	ADJ
asir-36063	64	15	programming	programming	NOUN
asir-36063	64	16	to	to	PART
asir-36063	64	17	solve	solve	VERB
asir-36063	64	18	the	the	DET
asir-36063	64	19	np	np	NOUN
asir-36063	64	20	-	-	PUNCT
asir-36063	64	21	hardness	hardness	NOUN
asir-36063	64	22	:	:	PUNCT
asir-36063	64	23	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max⁡⁡⁡⁡⟨	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max⁡⁡⁡⁡⟨	PROPN
asir-36063	64	24	�	�	PROPN
asir-36063	64	25	̃	̃	PROPN
asir-36063	64	26	�	�	PROPN
asir-36063	64	27	,	,	PUNCT
asir-36063	64	28	𝑿⟩	𝑿⟩	PROPN
asir-36063	65	1	−	−	PROPN
asir-36063	65	2	𝜆⟨𝑱2𝑛	𝜆⟨𝑱2𝑛	PROPN
asir-36063	65	3	,	,	PUNCT
asir-36063	65	4	𝑿⟩	𝑿⟩	PROPN
asir-36063	65	5	s.t	s.t	PROPN
asir-36063	65	6	.	.	PROPN
asir-36063	66	1	⁡𝑋𝑖𝑖	⁡𝑋𝑖𝑖	ADP
asir-36063	66	2	=	=	PUNCT
asir-36063	66	3	1,1	1,1	NUM
asir-36063	66	4	≤	≤	NUM
asir-36063	66	5	𝑖	𝑖	PRON
asir-36063	66	6	≤	≤	NOUN
asir-36063	66	7	2𝑛⁡⁡⁡⁡⁡⁡	2𝑛⁡⁡⁡⁡⁡⁡	NUM
asir-36063	66	8	⁡⁡⁡𝑿	⁡⁡⁡𝑿	SYM
asir-36063	66	9	≽	≽	NOUN
asir-36063	66	10	0	0	NUM
asir-36063	66	11	⁡𝜆	⁡𝜆	NOUN
asir-36063	66	12	>	>	X
asir-36063	66	13	0	0	PUNCT
asir-36063	67	1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.4	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.4	ADJ
asir-36063	67	2	）	）	PUNCT
asir-36063	67	3	(	(	PUNCT
asir-36063	67	4	2.4	2.4	NUM
asir-36063	67	5	)	)	PUNCT
asir-36063	67	6	is	be	AUX
asir-36063	67	7	a	a	DET
asir-36063	67	8	convex	convex	ADJ
asir-36063	67	9	relaxation	relaxation	NOUN
asir-36063	67	10	of	of	ADP
asir-36063	67	11	(	(	PUNCT
asir-36063	67	12	2.3	2.3	NUM
asir-36063	67	13	)	)	PUNCT
asir-36063	67	14	.	.	PUNCT
asir-36063	68	1	in	in	ADP
asir-36063	68	2	order	order	NOUN
asir-36063	68	3	to	to	PART
asir-36063	68	4	recover	recover	VERB
asir-36063	68	5	the	the	DET
asir-36063	68	6	communities	community	NOUN
asir-36063	68	7	in	in	ADP
asir-36063	68	8	the	the	DET
asir-36063	68	9	graph	graph	NOUN
asir-36063	68	10	,	,	PUNCT
asir-36063	68	11	we	we	PRON
asir-36063	68	12	intend	intend	VERB
asir-36063	68	13	to	to	PART
asir-36063	68	14	maximize	maximize	VERB
asir-36063	68	15	the	the	DET
asir-36063	68	16	difference	difference	NOUN
asir-36063	68	17	between	between	ADP
asir-36063	68	18	the	the	DET
asir-36063	68	19	in	in	ADP
asir-36063	68	20	-	-	PUNCT
asir-36063	68	21	community	community	NOUN
asir-36063	68	22	degree	degree	NOUN
asir-36063	68	23	and	and	CCONJ
asir-36063	68	24	the	the	DET
asir-36063	68	25	cross	cross	ADJ
asir-36063	68	26	-	-	ADJ
asir-36063	68	27	community	community	ADJ
asir-36063	68	28	degree	degree	NOUN
asir-36063	68	29	in	in	ADP
asir-36063	68	30	rows	row	NOUN
asir-36063	68	31	and	and	CCONJ
asir-36063	68	32	columns	column	NOUN
asir-36063	68	33	respectively	respectively	ADV
asir-36063	68	34	.	.	PUNCT
asir-36063	69	1	however	however	ADV
asir-36063	69	2	,	,	PUNCT
asir-36063	69	3	we	we	PRON
asir-36063	69	4	do	do	AUX
asir-36063	69	5	n't	not	PART
asir-36063	69	6	want	want	VERB
asir-36063	69	7	𝒖	𝒖	NOUN
asir-36063	69	8	and	and	CCONJ
asir-36063	69	9	𝒗	𝒗	VERB
asir-36063	69	10	to	to	PART
asir-36063	69	11	be	be	AUX
asir-36063	69	12	too	too	ADV
asir-36063	69	13	close	close	ADJ
asir-36063	69	14	to	to	ADP
asir-36063	69	15	all	all	ADV
asir-36063	69	16	-	-	PUNCT
asir-36063	69	17	one	one	NUM
asir-36063	69	18	vector	vector	NOUN
asir-36063	69	19	or	or	CCONJ
asir-36063	69	20	all	all	ADV
asir-36063	69	21	-	-	PUNCT
asir-36063	69	22	negative	negative	ADJ
asir-36063	69	23	one	one	NUM
asir-36063	69	24	vector	vector	NOUN
asir-36063	69	25	.	.	PUNCT
asir-36063	70	1	so	so	ADV
asir-36063	70	2	we	we	PRON
asir-36063	70	3	will	will	AUX
asir-36063	70	4	take	take	VERB
asir-36063	70	5	𝜆	𝜆	NOUN
asir-36063	70	6	=	=	SYM
asir-36063	70	7	1	1	NUM
asir-36063	70	8	2	2	NUM
asir-36063	70	9	,	,	PUNCT
asir-36063	70	10	and	and	CCONJ
asir-36063	70	11	(	(	PUNCT
asir-36063	70	12	2.4	2.4	NUM
asir-36063	70	13	)	)	PUNCT
asir-36063	70	14	becomes	become	VERB
asir-36063	70	15	:	:	PUNCT
asir-36063	70	16	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡max	NUM
asir-36063	70	17	tr⁡((2	tr⁡((2	X
asir-36063	70	18	�	�	PROPN
asir-36063	70	19	̃	̃	PROPN
asir-36063	70	20	�	�	PROPN
asir-36063	70	21	−	−	NUM
asir-36063	70	22	𝑱2𝑛)𝑿	𝑱2𝑛)𝑿	NUM
asir-36063	70	23	)	)	PUNCT
asir-36063	70	24	s.t	s.t	PROPN
asir-36063	70	25	.	.	PUNCT
asir-36063	71	1	𝑋𝑖𝑖	𝑋𝑖𝑖	NOUN
asir-36063	71	2	=	=	SYM
asir-36063	71	3	1	1	NUM
asir-36063	71	4	𝑿	𝑿	PROPN
asir-36063	71	5	≽	≽	PROPN
asir-36063	71	6	0	0	NUM
asir-36063	71	7	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.5	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.5	PROPN
asir-36063	71	8	）	）	X
asir-36063	71	9	note	note	NOUN
asir-36063	71	10	that	that	SCONJ
asir-36063	71	11	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡2	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡2	ADJ
asir-36063	71	12	�	�	PROPN
asir-36063	71	13	̃	̃	PROPN
asir-36063	71	14	�	�	NOUN
asir-36063	71	15	−	−	NOUN
asir-36063	71	16	𝑱2𝑛	𝑱2𝑛	PROPN
asir-36063	71	17	=	=	PUNCT
asir-36063	72	1	[	[	PUNCT
asir-36063	72	2	−𝑱𝑛	−𝑱𝑛	ADV
asir-36063	72	3	𝑩𝑇	𝑩𝑇	PROPN
asir-36063	72	4	𝑩	𝑩	PROPN
asir-36063	72	5	−𝑱𝑛	−𝑱𝑛	ADV
asir-36063	72	6	]	]	PUNCT
asir-36063	73	1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵𝑖𝑗	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵𝑖𝑗	PROPN
asir-36063	73	2	=	=	SYM
asir-36063	73	3	{	{	PUNCT
asir-36063	73	4	1⁡	1⁡	PROPN
asir-36063	73	5	if	if	SCONJ
asir-36063	73	6	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	73	7	=	=	AUX
asir-36063	73	8	1	1	NUM
asir-36063	73	9	−1⁡	−1⁡	NUM
asir-36063	73	10	if	if	SCONJ
asir-36063	73	11	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	73	12	=	=	NOUN
asir-36063	73	13	0	0	NUM
asir-36063	73	14	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.6	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡（2.6	NOUN
asir-36063	73	15	）	）	NOUN
asir-36063	73	16	3	3	X
asir-36063	73	17	.	.	X
asir-36063	73	18	main	main	ADJ
asir-36063	73	19	results	result	NOUN
asir-36063	73	20	given	give	VERB
asir-36063	73	21	the	the	DET
asir-36063	73	22	the	the	DET
asir-36063	73	23	dsbm⁡(𝑛	dsbm⁡(𝑛	PROPN
asir-36063	73	24	,	,	PUNCT
asir-36063	73	25	𝑝	𝑝	PROPN
asir-36063	73	26	,	,	PUNCT
asir-36063	73	27	𝑞	𝑞	NOUN
asir-36063	73	28	)	)	PUNCT
asir-36063	73	29	defined	define	VERB
asir-36063	73	30	in	in	ADP
asir-36063	73	31	section	section	NOUN
asir-36063	73	32	2.1	2.1	NUM
asir-36063	73	33	,	,	PUNCT
asir-36063	73	34	in	in	ADP
asir-36063	73	35	the	the	DET
asir-36063	73	36	regime	regime	NOUN
asir-36063	73	37	𝑝	𝑝	PROPN
asir-36063	73	38	=	=	SYM
asir-36063	73	39	𝑎log⁡(𝑛)/𝑛	𝑎log⁡(𝑛)/𝑛	PROPN
asir-36063	73	40	,	,	PUNCT
asir-36063	73	41	𝑞	𝑞	X
asir-36063	73	42	=	=	SYM
asir-36063	73	43	𝑏log⁡(𝑛)/𝑛	𝑏log⁡(𝑛)/𝑛	PROPN
asir-36063	73	44	,	,	PUNCT
asir-36063	73	45	we	we	PRON
asir-36063	73	46	will	will	AUX
asir-36063	73	47	present	present	VERB
asir-36063	73	48	the	the	DET
asir-36063	73	49	main	main	ADJ
asir-36063	73	50	argument	argument	NOUN
asir-36063	73	51	that	that	SCONJ
asir-36063	73	52	√𝑎	√𝑎	ADV
asir-36063	73	53	−	−	PROPN
asir-36063	73	54	√𝑏	√𝑏	NOUN
asir-36063	73	55	=	=	SYM
asir-36063	73	56	√2	√2	NOUN
asir-36063	73	57	is	be	AUX
asir-36063	73	58	the	the	DET
asir-36063	73	59	information	information	NOUN
asir-36063	73	60	-	-	PUNCT
asir-36063	73	61	theoretical	theoretical	ADJ
asir-36063	73	62	threshold	threshold	NOUN
asir-36063	73	63	for	for	ADP
asir-36063	73	64	exact	exact	ADJ
asir-36063	73	65	recovery	recovery	NOUN
asir-36063	73	66	in	in	ADP
asir-36063	73	67	the	the	DET
asir-36063	73	68	dsbm	dsbm	NOUN
asir-36063	73	69	.	.	PUNCT
asir-36063	74	1	to	to	PART
asir-36063	74	2	be	be	AUX
asir-36063	74	3	more	more	ADV
asir-36063	74	4	specific	specific	ADJ
asir-36063	74	5	,	,	PUNCT
asir-36063	74	6	the	the	DET
asir-36063	74	7	argument	argument	NOUN
asir-36063	74	8	will	will	AUX
asir-36063	74	9	be	be	AUX
asir-36063	74	10	presented	present	VERB
asir-36063	74	11	from	from	ADP
asir-36063	74	12	two	two	NUM
asir-36063	74	13	perspectives	perspective	NOUN
asir-36063	74	14	,	,	PUNCT
asir-36063	74	15	namely	namely	ADV
asir-36063	74	16	the	the	DET
asir-36063	74	17	impossibility	impossibility	NOUN
asir-36063	74	18	part	part	NOUN
asir-36063	74	19	and	and	CCONJ
asir-36063	74	20	the	the	DET
asir-36063	74	21	achievability	achievability	NOUN
asir-36063	74	22	part	part	NOUN
asir-36063	74	23	.	.	PUNCT
asir-36063	75	1	in	in	ADP
asir-36063	75	2	the	the	DET
asir-36063	75	3	impossibility	impossibility	NOUN
asir-36063	75	4	part	part	NOUN
asir-36063	75	5	,	,	PUNCT
asir-36063	75	6	we	we	PRON
asir-36063	75	7	will	will	AUX
asir-36063	75	8	show	show	VERB
asir-36063	75	9	that	that	SCONJ
asir-36063	75	10	when	when	SCONJ
asir-36063	75	11	√𝑎	√𝑎	ADJ
asir-36063	75	12	−	−	NOUN
asir-36063	75	13	√𝑏	√𝑏	VERB
asir-36063	75	14	<	<	X
asir-36063	75	15	√2	√2	NOUN
asir-36063	75	16	,	,	PUNCT
asir-36063	75	17	even	even	ADV
asir-36063	75	18	the	the	DET
asir-36063	75	19	mle	mle	NOUN
asir-36063	75	20	fails	fail	VERB
asir-36063	75	21	to	to	PART
asir-36063	75	22	recover	recover	VERB
asir-36063	75	23	the	the	DET
asir-36063	75	24	correct	correct	ADJ
asir-36063	75	25	membership	membership	NOUN
asir-36063	75	26	of	of	ADP
asir-36063	75	27	each	each	DET
asir-36063	75	28	node	node	NOUN
asir-36063	75	29	.	.	PUNCT
asir-36063	76	1	in	in	ADP
asir-36063	76	2	the	the	DET
asir-36063	76	3	achievability	achievability	NOUN
asir-36063	76	4	part	part	NOUN
asir-36063	76	5	,	,	PUNCT
asir-36063	76	6	we	we	PRON
asir-36063	76	7	will	will	AUX
asir-36063	76	8	show	show	VERB
asir-36063	76	9	that	that	SCONJ
asir-36063	76	10	when	when	SCONJ
asir-36063	76	11	√𝑎	√𝑎	ADJ
asir-36063	76	12	−	−	PROPN
asir-36063	76	13	√𝑏	√𝑏	PUNCT
asir-36063	76	14	>	>	X
asir-36063	76	15	√2	√2	NOUN
asir-36063	76	16	the	the	DET
asir-36063	76	17	sdp	sdp	NOUN
asir-36063	76	18	relaxation	relaxation	NOUN
asir-36063	76	19	can	can	AUX
asir-36063	76	20	correctly	correctly	ADV
asir-36063	76	21	recover	recover	VERB
asir-36063	76	22	the	the	DET
asir-36063	76	23	membership	membership	NOUN
asir-36063	76	24	of	of	ADP
asir-36063	76	25	each	each	DET
asir-36063	76	26	node	node	NOUN
asir-36063	76	27	with	with	ADP
asir-36063	76	28	high	high	ADJ
asir-36063	76	29	probability	probability	NOUN
asir-36063	76	30	.	.	PUNCT
asir-36063	77	1	we	we	PRON
asir-36063	77	2	only	only	ADV
asir-36063	77	3	provide	provide	VERB
asir-36063	77	4	a	a	DET
asir-36063	77	5	proof	proof	NOUN
asir-36063	77	6	sketch	sketch	NOUN
asir-36063	77	7	in	in	ADP
asir-36063	77	8	this	this	DET
asir-36063	77	9	section	section	NOUN
asir-36063	77	10	and	and	CCONJ
asir-36063	77	11	the	the	DET
asir-36063	77	12	detailed	detailed	ADJ
asir-36063	77	13	proofs	proof	NOUN
asir-36063	77	14	are	be	AUX
asir-36063	77	15	deferred	defer	VERB
asir-36063	77	16	to	to	PART
asir-36063	77	17	section	section	VERB
asir-36063	77	18	a	a	PRON
asir-36063	77	19	and	and	CCONJ
asir-36063	77	20	b.	b.	PROPN
asir-36063	77	21	3.1	3.1	NUM
asir-36063	77	22	impossibility	impossibility	NOUN
asir-36063	77	23	our	our	PRON
asir-36063	77	24	goal	goal	NOUN
asir-36063	77	25	in	in	ADP
asir-36063	77	26	this	this	DET
asir-36063	77	27	section	section	NOUN
asir-36063	77	28	is	be	AUX
asir-36063	77	29	to	to	PART
asir-36063	77	30	provide	provide	VERB
asir-36063	77	31	a	a	DET
asir-36063	77	32	proof	proof	ADJ
asir-36063	77	33	sketch	sketch	NOUN
asir-36063	77	34	of	of	ADP
asir-36063	77	35	the	the	DET
asir-36063	77	36	condition	condition	NOUN
asir-36063	77	37	for	for	ADP
asir-36063	77	38	which	which	PRON
asir-36063	77	39	the	the	DET
asir-36063	77	40	mle	mle	PROPN
asir-36063	77	41	algorithm	algorithm	PROPN
asir-36063	77	42	fails	fail	VERB
asir-36063	77	43	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	77	44	applied	apply	VERB
asir-36063	77	45	science	science	NOUN
asir-36063	77	46	and	and	CCONJ
asir-36063	77	47	innovative	innovative	ADJ
asir-36063	77	48	research	research	NOUN
asir-36063	77	49	vol	vol	NOUN
asir-36063	77	50	.	.	PROPN
asir-36063	78	1	8	8	NUM
asir-36063	78	2	,	,	PUNCT
asir-36063	78	3	no	no	INTJ
asir-36063	78	4	.	.	NOUN
asir-36063	78	5	1	1	NUM
asir-36063	78	6	,	,	PUNCT
asir-36063	78	7	2024	2024	NUM
asir-36063	78	8	68	68	NUM
asir-36063	78	9	published	publish	VERB
asir-36063	78	10	by	by	ADP
asir-36063	78	11	scholink	scholink	PROPN
asir-36063	78	12	inc	inc	PROPN
asir-36063	78	13	.	.	PROPN
asir-36063	78	14	to	to	PART
asir-36063	78	15	recover	recover	VERB
asir-36063	78	16	the	the	DET
asir-36063	78	17	communities	community	NOUN
asir-36063	78	18	in	in	ADP
asir-36063	78	19	dsbm	dsbm	ADJ
asir-36063	78	20	,	,	PUNCT
asir-36063	78	21	we	we	PRON
asir-36063	78	22	first	first	ADV
asir-36063	78	23	introduce	introduce	VERB
asir-36063	78	24	the	the	DET
asir-36063	78	25	concept	concept	NOUN
asir-36063	78	26	of	of	ADP
asir-36063	78	27	bad	bad	ADJ
asir-36063	78	28	vertices	vertex	NOUN
asir-36063	78	29	which	which	PRON
asir-36063	78	30	is	be	AUX
asir-36063	78	31	defined	define	VERB
asir-36063	78	32	with	with	ADP
asir-36063	78	33	respect	respect	NOUN
asir-36063	78	34	to	to	ADP
asir-36063	78	35	the	the	DET
asir-36063	78	36	connectedness	connectedness	NOUN
asir-36063	78	37	of	of	ADP
asir-36063	78	38	each	each	DET
asir-36063	78	39	node	node	NOUN
asir-36063	78	40	.	.	PUNCT
asir-36063	79	1	then	then	ADV
asir-36063	79	2	using	use	VERB
asir-36063	79	3	this	this	DET
asir-36063	79	4	concept	concept	NOUN
asir-36063	79	5	of	of	ADP
asir-36063	79	6	bad	bad	ADJ
asir-36063	79	7	vertices	vertex	NOUN
asir-36063	79	8	,	,	PUNCT
asir-36063	79	9	we	we	PRON
asir-36063	79	10	will	will	AUX
asir-36063	79	11	find	find	VERB
asir-36063	79	12	under	under	ADP
asir-36063	79	13	what	what	DET
asir-36063	79	14	condition	condition	NOUN
asir-36063	79	15	this	this	DET
asir-36063	79	16	bad	bad	ADJ
asir-36063	79	17	vertex	vertex	NOUN
asir-36063	79	18	definitely	definitely	ADV
asir-36063	79	19	exists	exist	VERB
asir-36063	79	20	.	.	PUNCT
asir-36063	80	1	therefore	therefore	ADV
asir-36063	80	2	,	,	PUNCT
asir-36063	80	3	when	when	SCONJ
asir-36063	80	4	this	this	DET
asir-36063	80	5	condition	condition	NOUN
asir-36063	80	6	is	be	AUX
asir-36063	80	7	met	meet	VERB
asir-36063	80	8	,	,	PUNCT
asir-36063	80	9	the	the	DET
asir-36063	80	10	mle	mle	NOUN
asir-36063	80	11	will	will	AUX
asir-36063	80	12	fail	fail	VERB
asir-36063	80	13	.	.	PUNCT
asir-36063	81	1	theorem	theorem	VERB
asir-36063	81	2	3.1	3.1	NUM
asir-36063	81	3	.	.	PUNCT
asir-36063	82	1	let	let	VERB
asir-36063	82	2	𝐺	𝐺	PRON
asir-36063	82	3	be	be	AUX
asir-36063	82	4	a	a	DET
asir-36063	82	5	graph	graph	NOUN
asir-36063	82	6	drawn	draw	VERB
asir-36063	82	7	from	from	ADP
asir-36063	82	8	dsbm⁡(𝑛	dsbm⁡(𝑛	PROPN
asir-36063	82	9	,	,	PUNCT
asir-36063	82	10	𝑝	𝑝	PROPN
asir-36063	82	11	,	,	PUNCT
asir-36063	82	12	𝑞	𝑞	NOUN
asir-36063	82	13	)	)	PUNCT
asir-36063	82	14	,	,	PUNCT
asir-36063	82	15	let	let	VERB
asir-36063	82	16	𝑝	𝑝	NOUN
asir-36063	82	17	=	=	PUNCT
asir-36063	82	18	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	82	19	)	)	PUNCT
asir-36063	82	20	𝑛	𝑛	NOUN
asir-36063	82	21	and	and	CCONJ
asir-36063	82	22	𝑞	𝑞	X
asir-36063	82	23	=	=	PUNCT
asir-36063	82	24	𝑏log⁡(𝑛	𝑏log⁡(𝑛	PROPN
asir-36063	82	25	)	)	PUNCT
asir-36063	82	26	𝑛	𝑛	NOUN
asir-36063	82	27	,	,	PUNCT
asir-36063	82	28	then	then	ADV
asir-36063	82	29	exact	exact	ADJ
asir-36063	82	30	recover	recover	NOUN
asir-36063	82	31	is	be	AUX
asir-36063	82	32	impossible	impossible	ADJ
asir-36063	82	33	if	if	SCONJ
asir-36063	82	34	√𝑎	√𝑎	ADJ
asir-36063	82	35	−	−	NOUN
asir-36063	82	36	√𝑏	√𝑏	NOUN
asir-36063	82	37	<	<	X
asir-36063	82	38	√2	√2	X
asir-36063	82	39	(	(	PUNCT
asir-36063	82	40	3.1	3.1	NUM
asir-36063	82	41	)	)	PUNCT
asir-36063	82	42	the	the	DET
asir-36063	82	43	following	follow	VERB
asir-36063	82	44	steps	step	NOUN
asir-36063	82	45	are	be	AUX
asir-36063	82	46	the	the	DET
asir-36063	82	47	proof	proof	NOUN
asir-36063	82	48	sketch	sketch	VERB
asir-36063	82	49	to	to	ADP
asir-36063	82	50	the	the	DET
asir-36063	82	51	above	above	ADJ
asir-36063	82	52	main	main	ADJ
asir-36063	82	53	theorem	theorem	NOUN
asir-36063	82	54	.	.	PUNCT
asir-36063	83	1	definition	definition	NOUN
asir-36063	83	2	3.1	3.1	NUM
asir-36063	83	3	.	.	PUNCT
asir-36063	84	1	we	we	PRON
asir-36063	84	2	define	define	VERB
asir-36063	84	3	the	the	DET
asir-36063	84	4	likelihood	likelihood	NOUN
asir-36063	84	5	function	function	NOUN
asir-36063	84	6	of	of	ADP
asir-36063	84	7	the	the	DET
asir-36063	84	8	dsbm	dsbm	NOUN
asir-36063	84	9	as	as	ADP
asir-36063	84	10	:	:	PUNCT
asir-36063	84	11	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	84	12	,	,	PUNCT
asir-36063	84	13	𝑦	𝑦	X
asir-36063	84	14	)	)	PUNCT
asir-36063	84	15	=	=	SYM
asir-36063	84	16	∏	∏	PROPN
asir-36063	84	17	 	 	SPACE
asir-36063	84	18	𝑖,𝑗∈[𝑛]2	𝑖,𝑗∈[𝑛]2	PUNCT
asir-36063	84	19	𝑃𝑖,𝑗	𝑃𝑖,𝑗	PROPN
asir-36063	84	20	𝐴𝑖,𝑗(1	𝐴𝑖,𝑗(1	NOUN
asir-36063	84	21	−	−	PROPN
asir-36063	84	22	𝑃𝑖𝑗	𝑃𝑖𝑗	PROPN
asir-36063	84	23	)	)	PUNCT
asir-36063	84	24	1−𝐴𝑖,𝑗	1−𝐴𝑖,𝑗	PROPN
asir-36063	84	25	(	(	PUNCT
asir-36063	84	26	3.2	3.2	NUM
asir-36063	84	27	)	)	PUNCT
asir-36063	84	28	where	where	SCONJ
asir-36063	84	29	𝑨	𝑨	PROPN
asir-36063	84	30	is	be	AUX
asir-36063	84	31	the	the	DET
asir-36063	84	32	adjacency	adjacency	NOUN
asir-36063	84	33	matrix	matrix	NOUN
asir-36063	84	34	and	and	CCONJ
asir-36063	84	35	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑷	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑷	NOUN
asir-36063	84	36	=	=	PUNCT
asir-36063	85	1	[	[	PUNCT
asir-36063	85	2	𝑝𝑱𝑛/2	𝑝𝑱𝑛/2	X
asir-36063	85	3	𝑞𝑱𝑛/2	𝑞𝑱𝑛/2	X
asir-36063	85	4	𝑞𝑱𝑛/2	𝑞𝑱𝑛/2	X
asir-36063	85	5	𝑝𝑱𝑛/2	𝑝𝑱𝑛/2	X
asir-36063	85	6	]	]	PUNCT
asir-36063	85	7	.	.	PUNCT
asir-36063	86	1	definition	definition	NOUN
asir-36063	86	2	3.2	3.2	NUM
asir-36063	86	3	.	.	PUNCT
asir-36063	87	1	we	we	PRON
asir-36063	87	2	define	define	VERB
asir-36063	87	3	the	the	DET
asir-36063	87	4	degree	degree	NOUN
asir-36063	87	5	matrices	matrix	NOUN
asir-36063	87	6	for	for	ADP
asir-36063	87	7	dsbm	dsbm	ADJ
asir-36063	87	8	as	as	ADP
asir-36063	87	9	the	the	DET
asir-36063	87	10	following	following	NOUN
asir-36063	87	11	:	:	PUNCT
asir-36063	87	12	(	(	PUNCT
asir-36063	87	13	𝑫𝑅	𝑫𝑅	PROPN
asir-36063	87	14	+	+	NOUN
asir-36063	87	15	)	)	PUNCT
asir-36063	87	16	𝑖𝑖	𝑖𝑖	ADP
asir-36063	87	17	:	:	PUNCT
asir-36063	87	18	=	=	SYM
asir-36063	87	19	{	{	PUNCT
asir-36063	87	20	∑	∑	PROPN
asir-36063	87	21	  	  	SPACE
asir-36063	87	22	𝑛/2	𝑛/2	PRON
asir-36063	87	23	𝑗=1	𝑗=1	PROPN
asir-36063	87	24	 	 	SPACE
asir-36063	87	25	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	87	26	⁡𝑖	⁡𝑖	VERB
asir-36063	87	27	∈	∈	PROPN
asir-36063	88	1	[	[	X
asir-36063	88	2	1	1	NUM
asir-36063	88	3	,	,	PUNCT
asir-36063	88	4	𝑛	𝑛	DET
asir-36063	88	5	2	2	NUM
asir-36063	88	6	]	]	PUNCT
asir-36063	88	7	∑	∑	PART
asir-36063	88	8	 	 	SPACE
asir-36063	88	9	𝑛	𝑛	PROPN
asir-36063	88	10	𝑗=𝑛/2	𝑗=𝑛/2	PROPN
asir-36063	89	1	+	+	ADJ
asir-36063	89	2	1	1	NUM
asir-36063	89	3	 	 	SPACE
asir-36063	89	4	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	89	5	⁡𝑖	⁡𝑖	VERB
asir-36063	89	6	∈	∈	PROPN
asir-36063	89	7	[	[	PUNCT
asir-36063	89	8	𝑛	𝑛	PROPN
asir-36063	89	9	2	2	NUM
asir-36063	89	10	+	+	NUM
asir-36063	89	11	1	1	NUM
asir-36063	89	12	,	,	PUNCT
asir-36063	89	13	𝑛	𝑛	NOUN
asir-36063	89	14	]	]	X
asir-36063	89	15	(	(	PUNCT
asir-36063	89	16	𝑫𝑅	𝑫𝑅	PROPN
asir-36063	89	17	−)𝑖𝑖	−)𝑖𝑖	NOUN
asir-36063	89	18	:	:	PUNCT
asir-36063	89	19	=	=	X
asir-36063	89	20	{	{	PUNCT
asir-36063	89	21	∑	∑	PROPN
asir-36063	89	22	 	 	SPACE
asir-36063	89	23	𝑛	𝑛	PROPN
asir-36063	89	24	𝑗=𝑛/2	𝑗=𝑛/2	PROPN
asir-36063	89	25	+	+	ADJ
asir-36063	89	26	1	1	NUM
asir-36063	89	27	 	 	SPACE
asir-36063	89	28	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	89	29	⁡𝑖	⁡𝑖	VERB
asir-36063	89	30	∈	∈	PROPN
asir-36063	90	1	[	[	X
asir-36063	90	2	1	1	NUM
asir-36063	90	3	,	,	PUNCT
asir-36063	90	4	𝑛	𝑛	DET
asir-36063	90	5	2	2	NUM
asir-36063	90	6	]	]	PUNCT
asir-36063	90	7	∑	∑	PART
asir-36063	90	8	  	  	SPACE
asir-36063	90	9	𝑛	𝑛	PRON
asir-36063	90	10	2	2	NUM
asir-36063	90	11	𝑗=1	𝑗=1	PART
asir-36063	90	12	 	 	SPACE
asir-36063	90	13	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	90	14	⁡𝑖	⁡𝑖	VERB
asir-36063	90	15	∈	∈	PROPN
asir-36063	90	16	[	[	PUNCT
asir-36063	90	17	𝑛	𝑛	PROPN
asir-36063	90	18	2	2	NUM
asir-36063	90	19	+	+	NUM
asir-36063	90	20	1	1	NUM
asir-36063	90	21	,	,	PUNCT
asir-36063	90	22	𝑛	𝑛	NOUN
asir-36063	90	23	]	]	X
asir-36063	90	24	(	(	PUNCT
asir-36063	90	25	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	90	26	+	+	NOUN
asir-36063	90	27	)	)	PUNCT
asir-36063	90	28	𝑖𝑖	𝑖𝑖	ADP
asir-36063	90	29	:	:	PUNCT
asir-36063	90	30	=	=	SYM
asir-36063	90	31	{	{	PUNCT
asir-36063	90	32	∑	∑	PROPN
asir-36063	90	33	  	  	SPACE
asir-36063	90	34	𝑛/2	𝑛/2	X
asir-36063	90	35	𝑖=1	𝑖=1	PUNCT
asir-36063	90	36	 	 	SPACE
asir-36063	91	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	91	2	⁡𝑖	⁡𝑖	VERB
asir-36063	91	3	∈	∈	PROPN
asir-36063	92	1	[	[	X
asir-36063	92	2	1	1	NUM
asir-36063	92	3	,	,	PUNCT
asir-36063	92	4	𝑛	𝑛	DET
asir-36063	92	5	2	2	NUM
asir-36063	92	6	]	]	PUNCT
asir-36063	92	7	∑	∑	PART
asir-36063	92	8	 	 	SPACE
asir-36063	92	9	𝑛	𝑛	PROPN
asir-36063	92	10	𝑖=	𝑖=	NOUN
asir-36063	92	11	𝑛	𝑛	VERB
asir-36063	92	12	2	2	NUM
asir-36063	92	13	+1	+1	NOUN
asir-36063	92	14	 	 	SPACE
asir-36063	92	15	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	92	16	⁡𝑖	⁡𝑖	VERB
asir-36063	92	17	∈	∈	PROPN
asir-36063	93	1	[	[	X
asir-36063	93	2	𝑛/2	𝑛/2	X
asir-36063	93	3	+	+	NOUN
asir-36063	93	4	1	1	NUM
asir-36063	93	5	,	,	PUNCT
asir-36063	93	6	𝑛	𝑛	NOUN
asir-36063	93	7	]	]	X
asir-36063	93	8	(	(	PUNCT
asir-36063	93	9	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	93	10	−)𝑖𝑖	−)𝑖𝑖	NOUN
asir-36063	93	11	:	:	PUNCT
asir-36063	93	12	=	=	X
asir-36063	93	13	{	{	PUNCT
asir-36063	93	14	∑	∑	PROPN
asir-36063	93	15	 	 	SPACE
asir-36063	93	16	𝑛	𝑛	PROPN
asir-36063	93	17	𝑖=	𝑖=	NOUN
asir-36063	93	18	𝑛	𝑛	VERB
asir-36063	93	19	2	2	NUM
asir-36063	93	20	+1	+1	NOUN
asir-36063	93	21	 	 	SPACE
asir-36063	93	22	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	93	23	⁡𝑖	⁡𝑖	VERB
asir-36063	93	24	∈	∈	PROPN
asir-36063	94	1	[	[	X
asir-36063	94	2	1	1	NUM
asir-36063	94	3	,	,	PUNCT
asir-36063	94	4	𝑛	𝑛	DET
asir-36063	94	5	2	2	NUM
asir-36063	94	6	]	]	PUNCT
asir-36063	94	7	∑	∑	PART
asir-36063	94	8	  	  	SPACE
asir-36063	94	9	𝑛	𝑛	DET
asir-36063	94	10	2	2	NUM
asir-36063	94	11	𝑖=1	𝑖=1	PUNCT
asir-36063	94	12	 	 	SPACE
asir-36063	94	13	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
asir-36063	94	14	⁡𝑖	⁡𝑖	VERB
asir-36063	94	15	∈	∈	PROPN
asir-36063	94	16	[	[	PUNCT
asir-36063	94	17	𝑛	𝑛	PROPN
asir-36063	94	18	2	2	NUM
asir-36063	94	19	+	+	NUM
asir-36063	94	20	1	1	NUM
asir-36063	94	21	,	,	PUNCT
asir-36063	94	22	𝑛	𝑛	NOUN
asir-36063	94	23	]	]	X
asir-36063	94	24	(	(	PUNCT
asir-36063	94	25	3.3	3.3	NUM
asir-36063	94	26	)	)	PUNCT
asir-36063	94	27	where	where	SCONJ
asir-36063	94	28	each	each	DET
asir-36063	94	29	entry	entry	NOUN
asir-36063	94	30	represents	represent	VERB
asir-36063	94	31	the	the	DET
asir-36063	94	32	number	number	NOUN
asir-36063	94	33	of	of	ADP
asir-36063	94	34	connections	connection	NOUN
asir-36063	94	35	that	that	PRON
asir-36063	94	36	satisfies	satisfy	VERB
asir-36063	94	37	the	the	DET
asir-36063	94	38	condition	condition	NOUN
asir-36063	94	39	described	describe	VERB
asir-36063	94	40	above	above	ADV
asir-36063	94	41	.	.	PUNCT
asir-36063	95	1	therefore	therefore	ADV
asir-36063	95	2	,	,	PUNCT
asir-36063	95	3	for	for	ADP
asir-36063	95	4	each	each	DET
asir-36063	95	5	vertex	vertex	NOUN
asir-36063	95	6	's	's	PART
asir-36063	95	7	connectedness	connectedness	NOUN
asir-36063	95	8	,	,	PUNCT
asir-36063	95	9	then	then	ADV
asir-36063	95	10	the	the	DET
asir-36063	95	11	in	in	ADP
asir-36063	95	12	-	-	PUNCT
asir-36063	95	13	degree	degree	NOUN
asir-36063	95	14	matrix	matrix	NOUN
asir-36063	95	15	and	and	CCONJ
asir-36063	95	16	the	the	DET
asir-36063	95	17	out	out	ADJ
asir-36063	95	18	-	-	PUNCT
asir-36063	95	19	degree	degree	NOUN
asir-36063	95	20	matrix	matrix	NOUN
asir-36063	95	21	for	for	ADP
asir-36063	95	22	dsbm	dsbm	ADJ
asir-36063	95	23	respectively	respectively	ADV
asir-36063	95	24	as	as	ADP
asir-36063	95	25	:	:	PUNCT
asir-36063	95	26	𝑫+	𝑫+	NUM
asir-36063	95	27	:	:	PUNCT
asir-36063	95	28	=	=	SYM
asir-36063	95	29	[	[	PUNCT
asir-36063	95	30	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	95	31	+	+	CCONJ
asir-36063	95	32	0	0	NUM
asir-36063	95	33	0	0	NUM
asir-36063	95	34	𝑫𝑅	𝑫𝑅	PROPN
asir-36063	95	35	+	+	SYM
asir-36063	95	36	]	]	X
asir-36063	95	37	⁡𝑫	⁡𝑫	ADP
asir-36063	95	38	−	−	NOUN
asir-36063	95	39	:	:	PUNCT
asir-36063	95	40	=	=	SYM
asir-36063	95	41	[	[	PUNCT
asir-36063	95	42	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	95	43	−	−	PROPN
asir-36063	95	44	0	0	NUM
asir-36063	95	45	0	0	NUM
asir-36063	95	46	𝑫𝑅	𝑫𝑅	PROPN
asir-36063	95	47	−	−	NOUN
asir-36063	95	48	]	]	PUNCT
asir-36063	95	49	for	for	ADP
asir-36063	95	50	each	each	DET
asir-36063	95	51	vertex	vertex	NOUN
asir-36063	95	52	's	's	PART
asir-36063	95	53	connectedness	connectedness	NOUN
asir-36063	95	54	,	,	PUNCT
asir-36063	95	55	we	we	PRON
asir-36063	95	56	can	can	AUX
asir-36063	95	57	use	use	VERB
asir-36063	95	58	𝑑(𝑖	𝑑(𝑖	NOUN
asir-36063	95	59	)	)	PUNCT
asir-36063	95	60	to	to	PART
asir-36063	95	61	represent	represent	VERB
asir-36063	95	62	the	the	DET
asir-36063	95	63	ith	ith	PROPN
asir-36063	95	64	vertex	vertex	NOUN
asir-36063	95	65	's	's	PART
asir-36063	95	66	degree	degree	NOUN
asir-36063	95	67	.	.	PUNCT
asir-36063	96	1	then	then	ADV
asir-36063	96	2	we	we	PRON
asir-36063	96	3	have	have	VERB
asir-36063	96	4	𝑑−(𝑖	𝑑−(𝑖	NOUN
asir-36063	96	5	)	)	PUNCT
asir-36063	96	6	to	to	PART
asir-36063	96	7	represent	represent	VERB
asir-36063	96	8	the	the	DET
asir-36063	96	9	degree	degree	NOUN
asir-36063	96	10	with	with	ADP
asir-36063	96	11	cross	cross	ADJ
asir-36063	96	12	-	-	ADJ
asir-36063	96	13	community	community	ADJ
asir-36063	96	14	vertex	vertex	NOUN
asir-36063	96	15	,	,	PUNCT
asir-36063	96	16	𝑑+(𝑖	𝑑+(𝑖	NOUN
asir-36063	96	17	)	)	PUNCT
asir-36063	96	18	to	to	PART
asir-36063	96	19	represent	represent	VERB
asir-36063	96	20	the	the	DET
asir-36063	96	21	degree	degree	NOUN
asir-36063	96	22	with	with	ADP
asir-36063	96	23	the	the	DET
asir-36063	96	24	same	same	ADJ
asir-36063	96	25	community	community	NOUN
asir-36063	96	26	.	.	PUNCT
asir-36063	97	1	𝑑𝑅(𝑖	𝑑𝑅(𝑖	VERB
asir-36063	97	2	)	)	PUNCT
asir-36063	97	3	to	to	PART
asir-36063	97	4	represent	represent	VERB
asir-36063	97	5	the	the	DET
asir-36063	97	6	degree	degree	NOUN
asir-36063	97	7	with	with	ADP
asir-36063	97	8	the	the	DET
asir-36063	97	9	row	row	NOUN
asir-36063	97	10	,	,	PUNCT
asir-36063	97	11	and	and	CCONJ
asir-36063	97	12	𝑑𝐶(𝑖	𝑑𝐶(𝑖	NUM
asir-36063	97	13	)	)	PUNCT
asir-36063	97	14	to	to	PART
asir-36063	97	15	represent	represent	VERB
asir-36063	97	16	the	the	DET
asir-36063	97	17	degree	degree	NOUN
asir-36063	97	18	with	with	ADP
asir-36063	97	19	the	the	DET
asir-36063	97	20	column	column	NOUN
asir-36063	97	21	.	.	PUNCT
asir-36063	98	1	the	the	DET
asir-36063	98	2	concept	concept	NOUN
asir-36063	98	3	of	of	ADP
asir-36063	98	4	bad	bad	ADJ
asir-36063	98	5	vertex	vertex	NOUN
asir-36063	98	6	and	and	CCONJ
asir-36063	98	7	bad	bad	ADJ
asir-36063	98	8	edges	edge	NOUN
asir-36063	98	9	is	be	AUX
asir-36063	98	10	essential	essential	ADJ
asir-36063	98	11	in	in	ADP
asir-36063	98	12	the	the	DET
asir-36063	98	13	proof	proof	NOUN
asir-36063	98	14	of	of	ADP
asir-36063	98	15	the	the	DET
asir-36063	98	16	impossibility	impossibility	NOUN
asir-36063	98	17	part	part	NOUN
asir-36063	98	18	.	.	PUNCT
asir-36063	99	1	it	it	PRON
asir-36063	99	2	can	can	AUX
asir-36063	99	3	be	be	AUX
asir-36063	99	4	verified	verify	VERB
asir-36063	99	5	that	that	SCONJ
asir-36063	99	6	the	the	DET
asir-36063	99	7	presence	presence	NOUN
asir-36063	99	8	of	of	ADP
asir-36063	99	9	bad	bad	ADJ
asir-36063	99	10	edges	edge	NOUN
asir-36063	99	11	and	and	CCONJ
asir-36063	99	12	bad	bad	ADJ
asir-36063	99	13	vertices	vertex	NOUN
asir-36063	99	14	implies	imply	VERB
asir-36063	99	15	the	the	DET
asir-36063	99	16	impossibility	impossibility	NOUN
asir-36063	99	17	of	of	ADP
asir-36063	99	18	exact	exact	ADJ
asir-36063	99	19	recovery	recovery	NOUN
asir-36063	99	20	and	and	CCONJ
asir-36063	99	21	the	the	DET
asir-36063	99	22	condition	condition	NOUN
asir-36063	99	23	√𝑎	√𝑎	ADV
asir-36063	99	24	−	−	NOUN
asir-36063	99	25	√𝑏	√𝑏	NOUN
asir-36063	99	26	<	<	X
asir-36063	99	27	√2	√2	NOUN
asir-36063	99	28	is	be	AUX
asir-36063	99	29	the	the	DET
asir-36063	99	30	sufficient	sufficient	ADJ
asir-36063	99	31	condition	condition	NOUN
asir-36063	99	32	for	for	ADP
asir-36063	99	33	the	the	DET
asir-36063	99	34	existence	existence	NOUN
asir-36063	99	35	of	of	ADP
asir-36063	99	36	bad	bad	ADJ
asir-36063	99	37	vertices	vertex	NOUN
asir-36063	99	38	.	.	PUNCT
asir-36063	100	1	definition	definition	NOUN
asir-36063	100	2	3.3	3.3	NUM
asir-36063	100	3	.	.	PUNCT
asir-36063	101	1	bad	bad	ADJ
asir-36063	101	2	vertices	vertex	NOUN
asir-36063	101	3	is	be	AUX
asir-36063	101	4	a	a	DET
asir-36063	101	5	type	type	NOUN
asir-36063	101	6	of	of	ADP
asir-36063	101	7	vertices	vertex	NOUN
asir-36063	101	8	pair	pair	NOUN
asir-36063	101	9	,	,	PUNCT
asir-36063	101	10	where	where	SCONJ
asir-36063	101	11	the	the	DET
asir-36063	101	12	two	two	NUM
asir-36063	101	13	vertices	vertex	NOUN
asir-36063	101	14	'	'	PART
asir-36063	101	15	community	community	NOUN
asir-36063	101	16	in	in	ADP
asir-36063	101	17	the	the	DET
asir-36063	101	18	pair	pair	NOUN
asir-36063	101	19	are	be	AUX
asir-36063	101	20	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	101	21	applied	apply	VERB
asir-36063	101	22	science	science	NOUN
asir-36063	101	23	and	and	CCONJ
asir-36063	101	24	innovative	innovative	ADJ
asir-36063	101	25	research	research	NOUN
asir-36063	101	26	vol	vol	NOUN
asir-36063	101	27	.	.	PROPN
asir-36063	101	28	8	8	NUM
asir-36063	101	29	,	,	PUNCT
asir-36063	101	30	no	no	INTJ
asir-36063	101	31	.	.	NOUN
asir-36063	101	32	1	1	NUM
asir-36063	101	33	,	,	PUNCT
asir-36063	101	34	2024	2024	NUM
asir-36063	101	35	69	69	NUM
asir-36063	101	36	published	publish	VERB
asir-36063	101	37	by	by	ADP
asir-36063	101	38	scholink	scholink	PROPN
asir-36063	101	39	inc	inc	PROPN
asir-36063	101	40	.	.	PROPN
asir-36063	101	41	swapped	swap	VERB
asir-36063	101	42	,	,	PUNCT
asir-36063	101	43	and	and	CCONJ
asir-36063	101	44	the	the	DET
asir-36063	101	45	mle	mle	NOUN
asir-36063	101	46	of	of	ADP
asir-36063	101	47	the	the	DET
asir-36063	101	48	swapped	swap	VERB
asir-36063	101	49	pair	pair	NOUN
asir-36063	101	50	is	be	AUX
asir-36063	101	51	larger	large	ADJ
asir-36063	101	52	than	than	ADP
asir-36063	101	53	the	the	DET
asir-36063	101	54	mle	mle	NOUN
asir-36063	101	55	of	of	ADP
asir-36063	101	56	the	the	DET
asir-36063	101	57	initial	initial	ADJ
asir-36063	101	58	pair	pair	NOUN
asir-36063	101	59	.	.	PUNCT
asir-36063	102	1	meaning	mean	VERB
asir-36063	102	2	that	that	SCONJ
asir-36063	102	3	after	after	ADP
asir-36063	102	4	the	the	DET
asir-36063	102	5	swap	swap	NOUN
asir-36063	102	6	,	,	PUNCT
asir-36063	102	7	vertices	vertex	NOUN
asir-36063	102	8	are	be	AUX
asir-36063	102	9	connected	connect	VERB
asir-36063	102	10	better	well	ADV
asir-36063	102	11	with	with	ADP
asir-36063	102	12	their	their	PRON
asir-36063	102	13	original	original	ADJ
asir-36063	102	14	community	community	NOUN
asir-36063	102	15	.	.	PUNCT
asir-36063	103	1	mathematically	mathematically	ADV
asir-36063	103	2	,	,	PUNCT
asir-36063	103	3	we	we	PRON
asir-36063	103	4	define	define	VERB
asir-36063	103	5	a	a	DET
asir-36063	103	6	pair	pair	NOUN
asir-36063	103	7	of	of	ADP
asir-36063	103	8	bad	bad	ADJ
asir-36063	103	9	vertices	vertex	NOUN
asir-36063	103	10	in	in	ADP
asir-36063	103	11	rows	row	NOUN
asir-36063	103	12	,	,	PUNCT
asir-36063	103	13	in	in	ADP
asir-36063	103	14	columns	column	NOUN
asir-36063	103	15	,	,	PUNCT
asir-36063	103	16	or	or	CCONJ
asir-36063	103	17	in	in	ADP
asir-36063	103	18	rows	row	NOUN
asir-36063	103	19	and	and	CCONJ
asir-36063	103	20	columns	column	NOUN
asir-36063	103	21	respectively	respectively	ADV
asir-36063	103	22	by	by	ADP
asir-36063	103	23	:	:	PUNCT
asir-36063	103	24	⁡𝐵𝑅(𝐺):=	⁡𝐵𝑅(𝐺):=	NUM
asir-36063	103	25	{	{	PUNCT
asir-36063	103	26	(	(	PUNCT
asir-36063	103	27	𝑢	𝑢	X
asir-36063	103	28	,	,	PUNCT
asir-36063	103	29	𝑣	𝑣	NOUN
asir-36063	103	30	):	):	PUNCT
asir-36063	103	31	𝑢	𝑢	PROPN
asir-36063	103	32	∈	∈	NOUN
asir-36063	103	33	𝐶1	𝐶1	PROPN
asir-36063	103	34	𝑅	𝑅	PROPN
asir-36063	103	35	,	,	PUNCT
asir-36063	103	36	𝑣	𝑣	PROPN
asir-36063	103	37	∈	∈	PROPN
asir-36063	103	38	𝐶2	𝐶2	PROPN
asir-36063	103	39	𝑅	𝑅	PROPN
asir-36063	103	40	,	,	PUNCT
asir-36063	103	41	𝐿(	𝐿(	PROPN
asir-36063	103	42	�	�	PROPN
asir-36063	103	43	̃	̃	PROPN
asir-36063	103	44	�	�	PROPN
asir-36063	103	45	,	,	PUNCT
asir-36063	103	46	𝑦	𝑦	NOUN
asir-36063	103	47	)	)	PUNCT
asir-36063	103	48	>	>	X
asir-36063	104	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	104	2	,	,	PUNCT
asir-36063	104	3	𝑦	𝑦	NOUN
asir-36063	104	4	)	)	PUNCT
asir-36063	104	5	}	}	PUNCT
asir-36063	104	6	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵𝐶(𝐺	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵𝐶(𝐺	PROPN
asir-36063	104	7	):	):	PUNCT
asir-36063	104	8	=	=	SYM
asir-36063	104	9	{	{	PUNCT
asir-36063	104	10	(	(	PUNCT
asir-36063	104	11	𝑢	𝑢	X
asir-36063	104	12	,	,	PUNCT
asir-36063	104	13	𝑣	𝑣	NOUN
asir-36063	104	14	):	):	PUNCT
asir-36063	104	15	𝑢	𝑢	PROPN
asir-36063	104	16	∈	∈	PROPN
asir-36063	104	17	𝐶1	𝐶1	PROPN
asir-36063	104	18	𝐶	𝐶	PROPN
asir-36063	104	19	,	,	PUNCT
asir-36063	104	20	𝑣	𝑣	PROPN
asir-36063	104	21	∈	∈	PROPN
asir-36063	104	22	𝐶2	𝐶2	PROPN
asir-36063	104	23	𝐶	𝐶	PROPN
asir-36063	104	24	,	,	PUNCT
asir-36063	104	25	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	104	26	,	,	PUNCT
asir-36063	104	27	�	�	PROPN
asir-36063	104	28	̃	̃	PROPN
asir-36063	104	29	�	�	PROPN
asir-36063	104	30	)	)	PUNCT
asir-36063	104	31	>	>	X
asir-36063	104	32	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	104	33	,	,	PUNCT
asir-36063	104	34	𝑦	𝑦	NOUN
asir-36063	104	35	)	)	PUNCT
asir-36063	104	36	}	}	PUNCT
asir-36063	104	37	𝐵𝑅,𝐶(𝐺	𝐵𝑅,𝐶(𝐺	NOUN
asir-36063	104	38	):	):	PUNCT
asir-36063	104	39	=	=	SYM
asir-36063	104	40	{	{	PUNCT
asir-36063	104	41	(	(	PUNCT
asir-36063	104	42	𝑢	𝑢	X
asir-36063	104	43	,	,	PUNCT
asir-36063	104	44	𝑣	𝑣	NOUN
asir-36063	104	45	):	):	PUNCT
asir-36063	104	46	𝑢	𝑢	PROPN
asir-36063	104	47	∈	∈	PROPN
asir-36063	104	48	𝐶1	𝐶1	NOUN
asir-36063	104	49	,	,	PUNCT
asir-36063	104	50	𝑣	𝑣	PROPN
asir-36063	104	51	∈	∈	PROPN
asir-36063	104	52	𝐶2	𝐶2	PROPN
asir-36063	104	53	,	,	PUNCT
asir-36063	104	54	𝐿(	𝐿(	PROPN
asir-36063	104	55	�	�	PROPN
asir-36063	104	56	̃	̃	PROPN
asir-36063	104	57	�	�	PROPN
asir-36063	104	58	,	,	PUNCT
asir-36063	104	59	�	�	PROPN
asir-36063	104	60	̃	̃	PROPN
asir-36063	104	61	�	�	PROPN
asir-36063	104	62	)	)	PUNCT
asir-36063	104	63	>	>	X
asir-36063	104	64	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	104	65	,	,	PUNCT
asir-36063	104	66	𝑦	𝑦	NOUN
asir-36063	104	67	)	)	PUNCT
asir-36063	104	68	}	}	PUNCT
asir-36063	104	69	(	(	PUNCT
asir-36063	104	70	3.4	3.4	NUM
asir-36063	104	71	)	)	PUNCT
asir-36063	104	72	then	then	ADV
asir-36063	104	73	we	we	PRON
asir-36063	104	74	are	be	AUX
asir-36063	104	75	trying	try	VERB
asir-36063	104	76	to	to	PART
asir-36063	104	77	prove	prove	VERB
asir-36063	104	78	that	that	SCONJ
asir-36063	104	79	for	for	ADP
asir-36063	104	80	√𝑎	√𝑎	ADJ
asir-36063	104	81	−	−	PROPN
asir-36063	104	82	√𝑏	√𝑏	NOUN
asir-36063	104	83	<	<	X
asir-36063	104	84	√2	√2	NOUN
asir-36063	104	85	,	,	PUNCT
asir-36063	104	86	there	there	PRON
asir-36063	104	87	exists	exist	VERB
asir-36063	104	88	at	at	ADP
asir-36063	104	89	least	least	ADV
asir-36063	104	90	one	one	NUM
asir-36063	104	91	bad	bad	ADJ
asir-36063	104	92	vertex	vertex	NOUN
asir-36063	104	93	in	in	ADP
asir-36063	104	94	rows	row	NOUN
asir-36063	104	95	or	or	CCONJ
asir-36063	104	96	columns	column	NOUN
asir-36063	104	97	,	,	PUNCT
asir-36063	104	98	which	which	PRON
asir-36063	104	99	would	would	AUX
asir-36063	104	100	result	result	VERB
asir-36063	104	101	in	in	ADP
asir-36063	104	102	max{𝐿(	max{𝐿(	PROPN
asir-36063	104	103	�	�	PROPN
asir-36063	104	104	̃	̃	PROPN
asir-36063	104	105	�	�	PROPN
asir-36063	104	106	,	,	PUNCT
asir-36063	104	107	𝑦	𝑦	NOUN
asir-36063	104	108	)	)	PUNCT
asir-36063	104	109	,	,	PUNCT
asir-36063	104	110	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	104	111	,	,	PUNCT
asir-36063	104	112	�	�	PROPN
asir-36063	104	113	̃	̃	PROPN
asir-36063	104	114	�	�	PROPN
asir-36063	104	115	)	)	PUNCT
asir-36063	104	116	,	,	PUNCT
asir-36063	104	117	𝐿(	𝐿(	PROPN
asir-36063	104	118	�	�	PROPN
asir-36063	104	119	̃	̃	PROPN
asir-36063	104	120	�	�	PROPN
asir-36063	104	121	,	,	PUNCT
asir-36063	104	122	�	�	PROPN
asir-36063	104	123	̃	̃	PROPN
asir-36063	104	124	�	�	PROPN
asir-36063	104	125	)	)	PUNCT
asir-36063	104	126	≥	≥	NOUN
asir-36063	104	127	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	104	128	,	,	PUNCT
asir-36063	104	129	𝑦	𝑦	NOUN
asir-36063	104	130	)	)	PUNCT
asir-36063	104	131	}	}	PUNCT
asir-36063	104	132	.	.	PUNCT
asir-36063	105	1	we	we	PRON
asir-36063	105	2	define	define	VERB
asir-36063	105	3	a	a	DET
asir-36063	105	4	pair	pair	NOUN
asir-36063	105	5	of	of	ADP
asir-36063	105	6	bad	bad	ADJ
asir-36063	105	7	vertex	vertex	NOUN
asir-36063	105	8	(	(	PUNCT
asir-36063	105	9	𝑢	𝑢	X
asir-36063	105	10	,	,	PUNCT
asir-36063	105	11	𝑣	𝑣	NOUN
asir-36063	105	12	)	)	PUNCT
asir-36063	105	13	,	,	PUNCT
asir-36063	105	14	the	the	DET
asir-36063	105	15	relationship	relationship	NOUN
asir-36063	105	16	between	between	ADP
asir-36063	105	17	degrees	degree	NOUN
asir-36063	105	18	can	can	AUX
asir-36063	105	19	be	be	AUX
asir-36063	105	20	inferred	infer	VERB
asir-36063	105	21	from	from	ADP
asir-36063	105	22	the	the	DET
asir-36063	105	23	following	follow	VERB
asir-36063	105	24	relationship	relationship	NOUN
asir-36063	105	25	between	between	ADP
asir-36063	105	26	mle	mle	PROPN
asir-36063	105	27	:	:	PUNCT
asir-36063	105	28	𝐿(	𝐿(	PROPN
asir-36063	105	29	�	�	PROPN
asir-36063	105	30	̃	̃	PROPN
asir-36063	105	31	�	�	PROPN
asir-36063	105	32	,	,	PUNCT
asir-36063	105	33	𝑦	𝑦	NOUN
asir-36063	105	34	)	)	PUNCT
asir-36063	105	35	>	>	X
asir-36063	106	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	106	2	,	,	PUNCT
asir-36063	106	3	𝑦	𝑦	NOUN
asir-36063	106	4	)	)	PUNCT
asir-36063	106	5	→	→	SYM
asir-36063	106	6	𝑑−	𝑑−	NOUN
asir-36063	106	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	106	8	)	)	PUNCT
asir-36063	107	1	+	+	NUM
asir-36063	107	2	𝑑−	𝑑−	NOUN
asir-36063	107	3	𝑅(𝑣	𝑅(𝑣	PUNCT
asir-36063	107	4	)	)	PUNCT
asir-36063	107	5	>	>	PUNCT
asir-36063	107	6	𝑑+	𝑑+	X
asir-36063	107	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	107	8	)	)	PUNCT
asir-36063	107	9	+	+	CCONJ
asir-36063	107	10	𝑑+	𝑑+	NOUN
asir-36063	107	11	𝑅(𝑣	𝑅(𝑣	NUM
asir-36063	107	12	)	)	PUNCT
asir-36063	107	13	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	107	14	,	,	PUNCT
asir-36063	107	15	�	�	PROPN
asir-36063	107	16	̃	̃	PROPN
asir-36063	107	17	�	�	PROPN
asir-36063	107	18	)	)	PUNCT
asir-36063	107	19	>	>	X
asir-36063	108	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	108	2	,	,	PUNCT
asir-36063	108	3	𝑦	𝑦	NOUN
asir-36063	108	4	)	)	PUNCT
asir-36063	108	5	→	→	SYM
asir-36063	108	6	𝑑−	𝑑−	NOUN
asir-36063	108	7	𝐶(𝑢	𝐶(𝑢	CCONJ
asir-36063	108	8	)	)	PUNCT
asir-36063	108	9	+	+	CCONJ
asir-36063	108	10	𝑑−	𝑑−	NOUN
asir-36063	108	11	𝐶(𝑣	𝐶(𝑣	NOUN
asir-36063	108	12	)	)	PUNCT
asir-36063	108	13	>	>	X
asir-36063	108	14	𝑑+	𝑑+	X
asir-36063	108	15	𝐶(𝑢	𝐶(𝑢	X
asir-36063	108	16	)	)	PUNCT
asir-36063	109	1	+	+	CCONJ
asir-36063	109	2	𝑑+	𝑑+	X
asir-36063	109	3	𝐶(𝑣)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(3.5	𝐶(𝑣)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(3.5	NOUN
asir-36063	109	4	)	)	PUNCT
asir-36063	109	5	𝐿(	𝐿(	PROPN
asir-36063	109	6	�	�	PROPN
asir-36063	109	7	̃	̃	PROPN
asir-36063	109	8	�	�	PROPN
asir-36063	109	9	,	,	PUNCT
asir-36063	109	10	�	�	PROPN
asir-36063	109	11	̃	̃	PROPN
asir-36063	109	12	�	�	PROPN
asir-36063	109	13	)	)	PUNCT
asir-36063	109	14	>	>	X
asir-36063	110	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	110	2	,	,	PUNCT
asir-36063	110	3	𝑦	𝑦	NOUN
asir-36063	110	4	)	)	PUNCT
asir-36063	110	5	→	→	SYM
asir-36063	110	6	𝑑−	𝑑−	NOUN
asir-36063	110	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	110	8	)	)	PUNCT
asir-36063	111	1	+	+	NUM
asir-36063	111	2	𝑑−	𝑑−	NOUN
asir-36063	111	3	𝑅(𝑣	𝑅(𝑣	PUNCT
asir-36063	111	4	)	)	PUNCT
asir-36063	112	1	+	+	NUM
asir-36063	112	2	𝑑−	𝑑−	NOUN
asir-36063	112	3	𝐶(𝑢	𝐶(𝑢	CCONJ
asir-36063	112	4	)	)	PUNCT
asir-36063	113	1	+	+	CCONJ
asir-36063	113	2	𝑑−	𝑑−	NOUN
asir-36063	113	3	𝐶(𝑣	𝐶(𝑣	NOUN
asir-36063	113	4	)	)	PUNCT
asir-36063	113	5	>	>	X
asir-36063	113	6	𝑑+	𝑑+	X
asir-36063	113	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	113	8	)	)	PUNCT
asir-36063	113	9	+	+	CCONJ
asir-36063	113	10	𝑑+	𝑑+	NOUN
asir-36063	113	11	𝑅(𝑣	𝑅(𝑣	NUM
asir-36063	113	12	)	)	PUNCT
asir-36063	113	13	+	+	NUM
asir-36063	113	14	𝑑+	𝑑+	X
asir-36063	113	15	𝐶(𝑢	𝐶(𝑢	NOUN
asir-36063	113	16	)	)	PUNCT
asir-36063	114	1	+	+	CCONJ
asir-36063	114	2	𝑑+	𝑑+	X
asir-36063	114	3	𝐶(𝑣	𝐶(𝑣	NOUN
asir-36063	114	4	)	)	PUNCT
asir-36063	114	5	since	since	SCONJ
asir-36063	114	6	if	if	SCONJ
asir-36063	114	7	𝐿(	𝐿(	PROPN
asir-36063	114	8	�	�	PROPN
asir-36063	114	9	̃	̃	PROPN
asir-36063	114	10	�	�	PROPN
asir-36063	114	11	,	,	PUNCT
asir-36063	114	12	𝑦	𝑦	NOUN
asir-36063	114	13	)	)	PUNCT
asir-36063	114	14	>	>	X
asir-36063	114	15	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	114	16	,	,	PUNCT
asir-36063	114	17	𝑦	𝑦	NOUN
asir-36063	114	18	)	)	PUNCT
asir-36063	114	19	,	,	PUNCT
asir-36063	114	20	then	then	ADV
asir-36063	114	21	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	114	22	,	,	PUNCT
asir-36063	114	23	�	�	PROPN
asir-36063	114	24	̃	̃	PROPN
asir-36063	114	25	�	�	PROPN
asir-36063	114	26	)	)	PUNCT
asir-36063	114	27	>	>	X
asir-36063	114	28	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	114	29	,	,	PUNCT
asir-36063	114	30	𝑦)or⁡𝐿(	𝑦)or⁡𝐿(	NOUN
asir-36063	114	31	�	�	PROPN
asir-36063	114	32	̃	̃	PROPN
asir-36063	114	33	�	�	PROPN
asir-36063	114	34	,	,	PUNCT
asir-36063	114	35	�	�	PROPN
asir-36063	114	36	̃	̃	PROPN
asir-36063	114	37	�	�	PROPN
asir-36063	114	38	)	)	PUNCT
asir-36063	114	39	>	>	X
asir-36063	115	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	115	2	,	,	PUNCT
asir-36063	115	3	𝑦	𝑦	NOUN
asir-36063	115	4	)	)	PUNCT
asir-36063	115	5	.	.	PUNCT
asir-36063	116	1	hence	hence	ADV
asir-36063	116	2	,	,	PUNCT
asir-36063	116	3	it	it	PRON
asir-36063	116	4	is	be	AUX
asir-36063	116	5	enough	enough	ADJ
asir-36063	116	6	to	to	PART
asir-36063	116	7	only	only	ADV
asir-36063	116	8	study	study	VERB
asir-36063	116	9	the	the	DET
asir-36063	116	10	bad	bad	ADJ
asir-36063	116	11	vertices	vertex	NOUN
asir-36063	116	12	in	in	ADP
asir-36063	116	13	rows	row	NOUN
asir-36063	116	14	or	or	CCONJ
asir-36063	116	15	in	in	ADP
asir-36063	116	16	columns	column	NOUN
asir-36063	116	17	.	.	PUNCT
asir-36063	117	1	definition	definition	NOUN
asir-36063	117	2	3.4	3.4	NUM
asir-36063	117	3	.	.	PUNCT
asir-36063	118	1	using	use	VERB
asir-36063	118	2	the	the	DET
asir-36063	118	3	concept	concept	NOUN
asir-36063	118	4	of	of	ADP
asir-36063	118	5	degree	degree	NOUN
asir-36063	118	6	,	,	PUNCT
asir-36063	118	7	we	we	PRON
asir-36063	118	8	can	can	AUX
asir-36063	118	9	define	define	VERB
asir-36063	118	10	a	a	DET
asir-36063	118	11	set	set	NOUN
asir-36063	118	12	of	of	ADP
asir-36063	118	13	bad	bad	ADJ
asir-36063	118	14	vertices	vertex	NOUN
asir-36063	118	15	in	in	ADP
asir-36063	118	16	rows	row	NOUN
asir-36063	118	17	:	:	PUNCT
asir-36063	118	18	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵𝑖	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵𝑖	NUM
asir-36063	118	19	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	118	20	)	)	PUNCT
asir-36063	118	21	=	=	PRON
asir-36063	118	22	{	{	PUNCT
asir-36063	118	23	∃𝑢	∃𝑢	PROPN
asir-36063	118	24	:	:	PUNCT
asir-36063	118	25	𝑢	𝑢	PROPN
asir-36063	118	26	∈	∈	PROPN
asir-36063	118	27	𝐶𝑖	𝐶𝑖	PROPN
asir-36063	118	28	𝑅	𝑅	PROPN
asir-36063	118	29	,	,	PUNCT
asir-36063	118	30	𝑑+	𝑑+	X
asir-36063	118	31	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	118	32	)	)	PUNCT
asir-36063	118	33	≤	≤	NUM
asir-36063	118	34	𝑑−	𝑑−	ADV
asir-36063	118	35	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	118	36	)	)	PUNCT
asir-36063	118	37	−	−	PROPN
asir-36063	118	38	1	1	NUM
asir-36063	118	39	}	}	PUNCT
asir-36063	118	40	,	,	PUNCT
asir-36063	118	41	𝑖	𝑖	SYM
asir-36063	118	42	=	=	SYM
asir-36063	118	43	1,2	1,2	NUM
asir-36063	118	44	(	(	PUNCT
asir-36063	118	45	3.6	3.6	NUM
asir-36063	118	46	)	)	PUNCT
asir-36063	118	47	where	where	SCONJ
asir-36063	118	48	𝑖	𝑖	PRON
asir-36063	118	49	represents	represent	VERB
asir-36063	118	50	the	the	DET
asir-36063	118	51	community	community	NOUN
asir-36063	118	52	.	.	PUNCT
asir-36063	119	1	lemma	lemma	PROPN
asir-36063	119	2	3.2	3.2	NUM
asir-36063	119	3	.	.	PUNCT
asir-36063	120	1	if	if	SCONJ
asir-36063	120	2	𝐵1	𝐵1	PROPN
asir-36063	120	3	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	120	4	)	)	PUNCT
asir-36063	120	5	is	be	AUX
asir-36063	120	6	non	non	ADJ
asir-36063	120	7	-	-	ADJ
asir-36063	120	8	empty	empty	ADJ
asir-36063	120	9	and	and	CCONJ
asir-36063	120	10	with	with	ADP
asir-36063	120	11	high	high	ADJ
asir-36063	120	12	probability	probability	NOUN
asir-36063	120	13	,	,	PUNCT
asir-36063	120	14	then	then	ADV
asir-36063	120	15	𝐵𝑅(𝐺	𝐵𝑅(𝐺	PROPN
asir-36063	120	16	)	)	PUNCT
asir-36063	120	17	is	be	AUX
asir-36063	120	18	non	non	ADJ
asir-36063	120	19	-	-	ADJ
asir-36063	120	20	empty	empty	ADJ
asir-36063	120	21	and	and	CCONJ
asir-36063	120	22	with	with	ADP
asir-36063	120	23	non	non	ADJ
asir-36063	120	24	-	-	ADJ
asir-36063	120	25	vanishing	vanishing	ADJ
asir-36063	120	26	probability	probability	NOUN
asir-36063	120	27	.	.	PUNCT
asir-36063	121	1	3.2	3.2	NUM
asir-36063	121	2	acheivability	acheivability	NOUN
asir-36063	121	3	recall	recall	VERB
asir-36063	121	4	that	that	SCONJ
asir-36063	121	5	our	our	PRON
asir-36063	121	6	goal	goal	NOUN
asir-36063	121	7	is	be	AUX
asir-36063	121	8	to	to	PART
asir-36063	121	9	show	show	VERB
asir-36063	121	10	that	that	SCONJ
asir-36063	121	11	√𝑎	√𝑎	ADJ
asir-36063	121	12	−	−	NOUN
asir-36063	121	13	√𝑏	√𝑏	NOUN
asir-36063	121	14	=	=	SYM
asir-36063	121	15	√2	√2	NOUN
asir-36063	121	16	is	be	AUX
asir-36063	121	17	the	the	DET
asir-36063	121	18	tight	tight	ADJ
asir-36063	121	19	threshold	threshold	NOUN
asir-36063	121	20	for	for	ADP
asir-36063	121	21	exact	exact	ADJ
asir-36063	121	22	recovery	recovery	NOUN
asir-36063	121	23	.	.	PUNCT
asir-36063	122	1	after	after	SCONJ
asir-36063	122	2	showing	show	VERB
asir-36063	122	3	that	that	SCONJ
asir-36063	122	4	when	when	SCONJ
asir-36063	122	5	√𝑎	√𝑎	ADJ
asir-36063	122	6	−	−	NOUN
asir-36063	122	7	√𝑏	√𝑏	VERB
asir-36063	122	8	<	<	X
asir-36063	122	9	√2	√2	NOUN
asir-36063	122	10	mle	mle	PROPN
asir-36063	122	11	fails	fail	VERB
asir-36063	122	12	to	to	PART
asir-36063	122	13	recover	recover	VERB
asir-36063	122	14	the	the	DET
asir-36063	122	15	communities	community	NOUN
asir-36063	122	16	,	,	PUNCT
asir-36063	122	17	we	we	PRON
asir-36063	122	18	will	will	AUX
asir-36063	122	19	show	show	VERB
asir-36063	122	20	that	that	SCONJ
asir-36063	122	21	the	the	DET
asir-36063	122	22	sdp	sdp	NOUN
asir-36063	122	23	relaxation	relaxation	NOUN
asir-36063	122	24	(	(	PUNCT
asir-36063	122	25	2.5	2.5	NUM
asir-36063	122	26	)	)	PUNCT
asir-36063	122	27	can	can	AUX
asir-36063	122	28	recover	recover	VERB
asir-36063	122	29	the	the	DET
asir-36063	122	30	communities	community	NOUN
asir-36063	122	31	otherwise	otherwise	ADV
asir-36063	122	32	.	.	PUNCT
asir-36063	123	1	theorem	theorem	VERB
asir-36063	123	2	3.3	3.3	NUM
asir-36063	123	3	.	.	PUNCT
asir-36063	124	1	let	let	VERB
asir-36063	124	2	𝐺	𝐺	PRON
asir-36063	124	3	be	be	AUX
asir-36063	124	4	a	a	DET
asir-36063	124	5	graph	graph	NOUN
asir-36063	124	6	drawn	draw	VERB
asir-36063	124	7	from	from	ADP
asir-36063	124	8	dsbm⁡(𝑛	dsbm⁡(𝑛	PROPN
asir-36063	124	9	,	,	PUNCT
asir-36063	124	10	𝑝	𝑝	PROPN
asir-36063	124	11	,	,	PUNCT
asir-36063	124	12	𝑞	𝑞	NOUN
asir-36063	124	13	)	)	PUNCT
asir-36063	124	14	,	,	PUNCT
asir-36063	124	15	let	let	VERB
asir-36063	124	16	𝑝	𝑝	NOUN
asir-36063	124	17	=	=	PUNCT
asir-36063	124	18	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	124	19	)	)	PUNCT
asir-36063	124	20	𝑛	𝑛	NOUN
asir-36063	124	21	and	and	CCONJ
asir-36063	124	22	𝑞	𝑞	X
asir-36063	124	23	=	=	PUNCT
asir-36063	124	24	𝑏log⁡(𝑛	𝑏log⁡(𝑛	PROPN
asir-36063	124	25	)	)	PUNCT
asir-36063	124	26	𝑛	𝑛	NOUN
asir-36063	124	27	,	,	PUNCT
asir-36063	124	28	if	if	SCONJ
asir-36063	124	29	√𝑎	√𝑎	ADJ
asir-36063	124	30	−	−	PROPN
asir-36063	124	31	√𝑏	√𝑏	NOUN
asir-36063	124	32	>	>	X
asir-36063	125	1	√2	√2	NOUN
asir-36063	125	2	,	,	PUNCT
asir-36063	125	3	(	(	PUNCT
asir-36063	125	4	3.7	3.7	NUM
asir-36063	125	5	)	)	PUNCT
asir-36063	125	6	then	then	ADV
asir-36063	125	7	the	the	DET
asir-36063	125	8	sdp	sdp	NOUN
asir-36063	125	9	relaxation	relaxation	NOUN
asir-36063	125	10	in	in	ADP
asir-36063	125	11	(	(	PUNCT
asir-36063	125	12	2.5	2.5	NUM
asir-36063	125	13	)	)	PUNCT
asir-36063	125	14	can	can	AUX
asir-36063	125	15	recover	recover	VERB
asir-36063	125	16	the	the	DET
asir-36063	125	17	communities	community	NOUN
asir-36063	125	18	with	with	ADP
asir-36063	125	19	high	high	ADJ
asir-36063	125	20	probability	probability	NOUN
asir-36063	125	21	.	.	PUNCT
asir-36063	126	1	without	without	ADP
asir-36063	126	2	loss	loss	NOUN
asir-36063	126	3	of	of	ADP
asir-36063	126	4	generality	generality	NOUN
asir-36063	126	5	,	,	PUNCT
asir-36063	126	6	we	we	PRON
asir-36063	126	7	suppose	suppose	VERB
asir-36063	126	8	that	that	SCONJ
asir-36063	126	9	the	the	DET
asir-36063	126	10	ground	ground	NOUN
asir-36063	126	11	-	-	PUNCT
asir-36063	126	12	truth	truth	NOUN
asir-36063	126	13	community	community	NOUN
asir-36063	126	14	indicator	indicator	NOUN
asir-36063	126	15	,	,	PUNCT
asir-36063	126	16	denoted	denote	VERB
asir-36063	126	17	by	by	ADP
asir-36063	126	18	𝒈	𝒈	PRON
asir-36063	126	19	is	be	AUX
asir-36063	126	20	(	(	PUNCT
asir-36063	126	21	𝟏𝑛/2	𝟏𝑛/2	PROPN
asir-36063	126	22	⊤	⊤	PROPN
asir-36063	126	23	,	,	PUNCT
asir-36063	126	24	−𝟏𝑛/2	−𝟏𝑛/2	PRON
asir-36063	126	25	⊤	⊤	NOUN
asir-36063	126	26	,	,	PUNCT
asir-36063	126	27	𝟏𝑛/2	𝟏𝑛/2	PROPN
asir-36063	126	28	⊤	⊤	PROPN
asir-36063	126	29	,	,	PUNCT
asir-36063	126	30	−𝟏𝑛/2	−𝟏𝑛/2	PRON
asir-36063	126	31	⊤	⊤	PROPN
asir-36063	126	32	)	)	PUNCT
asir-36063	126	33	⊤	⊤	NOUN
asir-36063	126	34	.	.	PUNCT
asir-36063	127	1	since	since	SCONJ
asir-36063	127	2	we	we	PRON
asir-36063	127	3	use	use	VERB
asir-36063	127	4	the	the	DET
asir-36063	127	5	notion	notion	NOUN
asir-36063	127	6	of	of	ADP
asir-36063	127	7	co	co	ADJ
asir-36063	127	8	-	-	ADJ
asir-36063	127	9	clustering	clustering	ADJ
asir-36063	127	10	introduced	introduce	VERB
asir-36063	127	11	section	section	NOUN
asir-36063	127	12	2.2	2.2	NUM
asir-36063	127	13	,	,	PUNCT
asir-36063	127	14	𝒈	𝒈	PROPN
asir-36063	127	15	is	be	AUX
asir-36063	127	16	the	the	DET
asir-36063	127	17	stack	stack	NOUN
asir-36063	127	18	of	of	ADP
asir-36063	127	19	the	the	DET
asir-36063	127	20	column	column	NOUN
asir-36063	127	21	communities	community	NOUN
asir-36063	127	22	and	and	CCONJ
asir-36063	127	23	the	the	DET
asir-36063	127	24	row	row	NOUN
asir-36063	127	25	communities	community	NOUN
asir-36063	127	26	.	.	PUNCT
asir-36063	128	1	then	then	ADV
asir-36063	128	2	we	we	PRON
asir-36063	128	3	claim	claim	VERB
asir-36063	128	4	that	that	SCONJ
asir-36063	128	5	when	when	SCONJ
asir-36063	128	6	√𝑎	√𝑎	ADJ
asir-36063	128	7	−	−	PROPN
asir-36063	128	8	√𝑏	√𝑏	X
asir-36063	128	9	>	>	X
asir-36063	128	10	√2	√2	NOUN
asir-36063	128	11	,	,	PUNCT
asir-36063	128	12	(	(	PUNCT
asir-36063	128	13	2.5	2.5	NUM
asir-36063	128	14	)	)	PUNCT
asir-36063	128	15	has	have	AUX
asir-36063	128	16	a	a	DET
asir-36063	128	17	unique	unique	ADJ
asir-36063	128	18	solution	solution	NOUN
asir-36063	128	19	equal	equal	ADJ
asir-36063	128	20	to	to	ADP
asir-36063	128	21	𝒈𝒈⊤.	𝒈𝒈⊤.	PROPN
asir-36063	128	22	the	the	DET
asir-36063	128	23	following	follow	VERB
asir-36063	128	24	lemma	lemma	PROPN
asir-36063	128	25	uses	use	VERB
asir-36063	128	26	a	a	DET
asir-36063	128	27	surrogate	surrogate	ADJ
asir-36063	128	28	𝚲	𝚲	NOUN
asir-36063	128	29	to	to	PART
asir-36063	128	30	quantify	quantify	VERB
asir-36063	128	31	the	the	DET
asir-36063	128	32	condition	condition	NOUN
asir-36063	128	33	under	under	ADP
asir-36063	128	34	which	which	PRON
asir-36063	128	35	𝒈𝒈⊤	𝒈𝒈⊤	ADV
asir-36063	128	36	is	be	AUX
asir-36063	128	37	the	the	DET
asir-36063	128	38	unique	unique	ADJ
asir-36063	128	39	solution	solution	NOUN
asir-36063	128	40	.	.	PUNCT
asir-36063	129	1	definition	definition	NOUN
asir-36063	129	2	3.5	3.5	NUM
asir-36063	129	3	.	.	PUNCT
asir-36063	130	1	given	give	VERB
asir-36063	130	2	a	a	DET
asir-36063	130	3	graph	graph	NOUN
asir-36063	130	4	𝐺	𝐺	NOUN
asir-36063	130	5	drawn	draw	VERB
asir-36063	130	6	from	from	ADP
asir-36063	130	7	dsbm	dsbm	ADJ
asir-36063	130	8	with	with	ADP
asir-36063	130	9	two	two	NUM
asir-36063	130	10	clusters	cluster	NOUN
asir-36063	130	11	,	,	PUNCT
asir-36063	130	12	we	we	PRON
asir-36063	130	13	can	can	AUX
asir-36063	130	14	have	have	AUX
asir-36063	130	15	:	:	PUNCT
asir-36063	130	16	γdsbm	γdsbm	VERB
asir-36063	130	17	=	=	PUNCT
asir-36063	130	18	𝑫+	𝑫+	PUNCT
asir-36063	131	1	−𝑫−	−𝑫−	PROPN
asir-36063	131	2	−	−	PROPN
asir-36063	131	3	𝑨∗	𝑨∗	X
asir-36063	131	4	(	(	PUNCT
asir-36063	131	5	3.8	3.8	NUM
asir-36063	131	6	)	)	PUNCT
asir-36063	131	7	lemma	lemma	PROPN
asir-36063	131	8	3.4	3.4	NUM
asir-36063	131	9	.	.	PUNCT
asir-36063	132	1	let	let	VERB
asir-36063	132	2	𝚲	𝚲	NOUN
asir-36063	132	3	=	=	SYM
asir-36063	132	4	2γdsbm	2γdsbm	NUM
asir-36063	133	1	+	+	CCONJ
asir-36063	133	2	𝑰2𝑛	𝑰2𝑛	PROPN
asir-36063	133	3	+	+	PUNCT
asir-36063	133	4	[	[	PUNCT
asir-36063	133	5	2𝑱𝑛	2𝑱𝑛	NUM
asir-36063	133	6	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	133	7	−	−	PROPN
asir-36063	133	8	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	133	9	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	133	10	−	−	PROPN
asir-36063	133	11	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	133	12	2𝑱𝑛	2𝑱𝑛	PROPN
asir-36063	133	13	]	]	PUNCT
asir-36063	133	14	,	,	PUNCT
asir-36063	133	15	if	if	SCONJ
asir-36063	133	16	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	133	17	applied	apply	VERB
asir-36063	133	18	science	science	NOUN
asir-36063	133	19	and	and	CCONJ
asir-36063	133	20	innovative	innovative	ADJ
asir-36063	133	21	research	research	NOUN
asir-36063	133	22	vol	vol	NOUN
asir-36063	133	23	.	.	PROPN
asir-36063	133	24	8	8	NUM
asir-36063	133	25	,	,	PUNCT
asir-36063	133	26	no	no	INTJ
asir-36063	133	27	.	.	NOUN
asir-36063	133	28	1	1	NUM
asir-36063	133	29	,	,	PUNCT
asir-36063	133	30	2024	2024	NUM
asir-36063	133	31	70	70	NUM
asir-36063	133	32	published	publish	VERB
asir-36063	133	33	by	by	ADP
asir-36063	133	34	scholink	scholink	PROPN
asir-36063	133	35	inc	inc	PROPN
asir-36063	133	36	.	.	PUNCT
asir-36063	134	1	𝚲	𝚲	PROPN
asir-36063	134	2	≽	≽	PROPN
asir-36063	134	3	0	0	NUM
asir-36063	134	4	,	,	PUNCT
asir-36063	134	5	⁡𝜆2(𝚲	⁡𝜆2(𝚲	PROPN
asir-36063	134	6	)	)	PUNCT
asir-36063	134	7	>	>	X
asir-36063	135	1	0	0	X
asir-36063	135	2	.	.	PUNCT
asir-36063	136	1	then	then	ADV
asir-36063	136	2	𝒈𝒈⊤	𝒈𝒈⊤	ADV
asir-36063	136	3	is	be	AUX
asir-36063	136	4	the	the	DET
asir-36063	136	5	unique	unique	ADJ
asir-36063	136	6	solution	solution	NOUN
asir-36063	136	7	to	to	ADP
asir-36063	136	8	the	the	DET
asir-36063	136	9	sdp	sdp	NOUN
asir-36063	136	10	relaxation	relaxation	NOUN
asir-36063	136	11	(	(	PUNCT
asir-36063	136	12	2.5	2.5	NUM
asir-36063	136	13	)	)	PUNCT
asir-36063	136	14	.	.	PUNCT
asir-36063	137	1	since	since	SCONJ
asir-36063	137	2	𝚲	𝚲	NOUN
asir-36063	137	3	is	be	AUX
asir-36063	137	4	random	random	ADJ
asir-36063	137	5	,	,	PUNCT
asir-36063	137	6	the	the	DET
asir-36063	137	7	second	second	ADJ
asir-36063	137	8	smallest	small	ADJ
asir-36063	137	9	eigenvalue	eigenvalue	NOUN
asir-36063	137	10	of	of	ADP
asir-36063	137	11	it	it	PRON
asir-36063	137	12	is	be	AUX
asir-36063	137	13	hard	hard	ADJ
asir-36063	137	14	to	to	PART
asir-36063	137	15	calculate	calculate	VERB
asir-36063	137	16	.	.	PUNCT
asir-36063	138	1	however	however	ADV
asir-36063	138	2	,	,	PUNCT
asir-36063	138	3	thanks	thank	NOUN
asir-36063	138	4	to	to	ADP
asir-36063	138	5	weyl	weyl	PROPN
asir-36063	138	6	's	's	PART
asir-36063	138	7	inequality	inequality	NOUN
asir-36063	138	8	,	,	PUNCT
asir-36063	138	9	we	we	PRON
asir-36063	138	10	can	can	AUX
asir-36063	138	11	estimate	estimate	VERB
asir-36063	138	12	it	it	PRON
asir-36063	138	13	using	use	VERB
asir-36063	138	14	its	its	PRON
asir-36063	138	15	population	population	NOUN
asir-36063	138	16	counterpart	counterpart	NOUN
asir-36063	138	17	,	,	PUNCT
asir-36063	138	18	i.e.	i.e.	X
asir-36063	138	19	,	,	PUNCT
asir-36063	138	20	the	the	DET
asir-36063	138	21	second	second	ADJ
asir-36063	138	22	smallest	small	ADJ
asir-36063	138	23	eigenvalue	eigenvalue	NOUN
asir-36063	138	24	of	of	ADP
asir-36063	138	25	𝔼𝚲.	𝔼𝚲.	PRON
asir-36063	138	26	the	the	DET
asir-36063	138	27	following	follow	VERB
asir-36063	138	28	lemma	lemma	PROPN
asir-36063	138	29	specifies	specify	VERB
asir-36063	138	30	the	the	DET
asir-36063	138	31	distance	distance	NOUN
asir-36063	138	32	between	between	ADP
asir-36063	138	33	the	the	DET
asir-36063	138	34	two	two	NUM
asir-36063	138	35	quantities	quantity	NOUN
asir-36063	138	36	.	.	PUNCT
asir-36063	139	1	lemma	lemma	PROPN
asir-36063	139	2	3.5	3.5	NUM
asir-36063	139	3	.	.	PUNCT
asir-36063	140	1	let	let	VERB
asir-36063	140	2	𝑛	𝑛	PRON
asir-36063	140	3	>	>	X
asir-36063	140	4	4	4	NUM
asir-36063	140	5	be	be	AUX
asir-36063	140	6	even	even	ADV
asir-36063	140	7	and	and	CCONJ
asir-36063	140	8	𝜆max(−γ𝑆𝐵𝑀	𝜆max(−γ𝑆𝐵𝑀	ADJ
asir-36063	140	9	+	+	NUM
asir-36063	140	10	𝔼[γ𝑆𝐵𝑀	𝔼[γ𝑆𝐵𝑀	PROPN
asir-36063	140	11	]	]	PUNCT
asir-36063	140	12	)	)	PUNCT
asir-36063	140	13	<	<	X
asir-36063	141	1	𝑛(𝑝	𝑛(𝑝	ADJ
asir-36063	141	2	−	−	PUNCT
asir-36063	141	3	𝑞	𝑞	NOUN
asir-36063	141	4	)	)	PUNCT
asir-36063	141	5	,	,	PUNCT
asir-36063	141	6	(	(	PUNCT
asir-36063	141	7	3.9	3.9	NUM
asir-36063	141	8	)	)	PUNCT
asir-36063	141	9	then	then	ADV
asir-36063	141	10	the	the	DET
asir-36063	141	11	sdp	sdp	NOUN
asir-36063	141	12	relaxation	relaxation	NOUN
asir-36063	141	13	for	for	ADP
asir-36063	141	14	dsbm	dsbm	ADJ
asir-36063	141	15	can	can	AUX
asir-36063	141	16	achieve	achieve	VERB
asir-36063	141	17	exact	exact	ADJ
asir-36063	141	18	recovery	recovery	NOUN
asir-36063	141	19	,	,	PUNCT
asir-36063	141	20	meaning	mean	VERB
asir-36063	141	21	that	that	SCONJ
asir-36063	141	22	gg⊤	gg⊤	VERB
asir-36063	141	23	is	be	AUX
asir-36063	141	24	the	the	DET
asir-36063	141	25	only	only	ADJ
asir-36063	141	26	solution	solution	NOUN
asir-36063	141	27	.	.	PUNCT
asir-36063	142	1	the	the	DET
asir-36063	142	2	following	follow	VERB
asir-36063	142	3	theorem	theorem	PROPN
asir-36063	142	4	finalizes	finalize	NOUN
asir-36063	142	5	the	the	DET
asir-36063	142	6	proof	proof	NOUN
asir-36063	142	7	in	in	ADP
asir-36063	142	8	this	this	DET
asir-36063	142	9	part	part	NOUN
asir-36063	142	10	by	by	ADP
asir-36063	142	11	connecting	connect	VERB
asir-36063	142	12	the	the	DET
asir-36063	142	13	degree	degree	NOUN
asir-36063	142	14	and	and	CCONJ
asir-36063	142	15	community	community	NOUN
asir-36063	142	16	detection	detection	NOUN
asir-36063	142	17	in	in	ADP
asir-36063	142	18	dsbm	dsbm	ADJ
asir-36063	142	19	.	.	PUNCT
asir-36063	143	1	theorem	theorem	VERB
asir-36063	143	2	3.6	3.6	NUM
asir-36063	143	3	.	.	PUNCT
asir-36063	144	1	let	let	VERB
asir-36063	144	2	𝑛	𝑛	PROPN
asir-36063	144	3	≥	≥	PRON
asir-36063	144	4	4	4	NUM
asir-36063	144	5	be	be	AUX
asir-36063	144	6	even	even	ADV
asir-36063	144	7	and	and	CCONJ
asir-36063	144	8	𝑮	𝑮	PROPN
asir-36063	144	9	be	be	VERB
asir-36063	144	10	a	a	DET
asir-36063	144	11	graph	graph	NOUN
asir-36063	144	12	of	of	ADP
asir-36063	144	13	directed	direct	VERB
asir-36063	144	14	stochastic	stochastic	ADJ
asir-36063	144	15	block	block	NOUN
asir-36063	144	16	model	model	NOUN
asir-36063	144	17	drawn	draw	VERB
asir-36063	144	18	from	from	ADP
asir-36063	144	19	𝒢(𝑛	𝒢(𝑛	NOUN
asir-36063	144	20	,	,	PUNCT
asir-36063	144	21	𝑝	𝑝	PROPN
asir-36063	144	22	,	,	PUNCT
asir-36063	144	23	𝑞	𝑞	NOUN
asir-36063	144	24	)	)	PUNCT
asir-36063	144	25	,	,	PUNCT
asir-36063	144	26	where	where	SCONJ
asir-36063	144	27	𝑝	𝑝	X
asir-36063	144	28	>	>	X
asir-36063	144	29	𝑞.	𝑞.	NOUN
asir-36063	144	30	only	only	ADV
asir-36063	144	31	when	when	SCONJ
asir-36063	144	32	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	144	33	)	)	PUNCT
asir-36063	144	34	𝑛	𝑛	ADP
asir-36063	144	35	<	<	X
asir-36063	144	36	𝑞	𝑞	X
asir-36063	144	37	<	<	X
asir-36063	144	38	𝑝	𝑝	X
asir-36063	144	39	<	<	X
asir-36063	144	40	1	1	NUM
asir-36063	144	41	2	2	NUM
asir-36063	144	42	,	,	PUNCT
asir-36063	144	43	and	and	CCONJ
asir-36063	144	44	for	for	ADP
asir-36063	144	45	some	some	DET
asir-36063	144	46	constant	constant	ADJ
asir-36063	144	47	𝑐	𝑐	PROPN
asir-36063	144	48	>	>	X
asir-36063	144	49	1	1	NUM
asir-36063	144	50	,	,	PUNCT
asir-36063	144	51	then	then	ADV
asir-36063	144	52	𝚫	𝚫	PROPN
asir-36063	144	53	>	>	X
asir-36063	144	54	0	0	NUM
asir-36063	144	55	such	such	ADJ
asir-36063	144	56	that	that	SCONJ
asir-36063	144	57	,	,	PUNCT
asir-36063	144	58	with	with	ADP
asir-36063	144	59	high	high	ADJ
asir-36063	144	60	probability	probability	NOUN
asir-36063	144	61	,	,	PUNCT
asir-36063	144	62	the	the	DET
asir-36063	144	63	following	follow	VERB
asir-36063	144	64	equation	equation	NOUN
asir-36063	144	65	holds	hold	VERB
asir-36063	144	66	:	:	PUNCT
asir-36063	144	67	if	if	SCONJ
asir-36063	144	68	min	min	NOUN
asir-36063	144	69	𝑖∈[2𝑛	𝑖∈[2𝑛	X
asir-36063	144	70	]	]	PUNCT
asir-36063	144	71	 	 	SPACE
asir-36063	144	72	(	(	PUNCT
asir-36063	144	73	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	144	74	+	+	CCONJ
asir-36063	144	75	−𝒟𝑖𝑖	−𝒟𝑖𝑖	PROPN
asir-36063	144	76	−	−	NOUN
asir-36063	144	77	)	)	PUNCT
asir-36063	144	78	≥	≥	NOUN
asir-36063	144	79	𝚫	𝚫	PROPN
asir-36063	144	80	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	144	81	)	)	PUNCT
asir-36063	144	82	𝔼[deg𝐶	𝔼[deg𝐶	NOUN
asir-36063	144	83	+	+	NOUN
asir-36063	144	84	⁡(𝑖	⁡(𝑖	NOUN
asir-36063	144	85	)	)	PUNCT
asir-36063	145	1	−	−	ADP
asir-36063	145	2	deg𝐶	deg𝐶	ADJ
asir-36063	145	3	−⁡(𝑖	−⁡(𝑖	PROPN
asir-36063	145	4	)	)	PUNCT
asir-36063	145	5	]	]	PUNCT
asir-36063	146	1	(	(	PUNCT
asir-36063	146	2	3.10	3.10	NUM
asir-36063	146	3	)	)	PUNCT
asir-36063	146	4	then	then	ADV
asir-36063	146	5	the	the	DET
asir-36063	146	6	semidefinite	semidefinite	PROPN
asir-36063	146	7	program	program	NOUN
asir-36063	146	8	achieve	achieve	VERB
asir-36063	146	9	exact	exact	ADJ
asir-36063	146	10	recovery	recovery	NOUN
asir-36063	146	11	.	.	PUNCT
asir-36063	147	1	now	now	ADV
asir-36063	147	2	we	we	PRON
asir-36063	147	3	have	have	VERB
asir-36063	147	4	the	the	DET
asir-36063	147	5	equation	equation	NOUN
asir-36063	147	6	represented	represent	VERB
asir-36063	147	7	by	by	ADP
asir-36063	147	8	degree	degree	NOUN
asir-36063	147	9	,	,	PUNCT
asir-36063	147	10	which	which	PRON
asir-36063	147	11	is	be	AUX
asir-36063	147	12	easier	easy	ADJ
asir-36063	147	13	and	and	CCONJ
asir-36063	147	14	more	more	ADV
asir-36063	147	15	straightforward	straightforward	ADJ
asir-36063	147	16	to	to	PART
asir-36063	147	17	solve	solve	VERB
asir-36063	147	18	than	than	ADP
asir-36063	147	19	the	the	DET
asir-36063	147	20	previous	previous	ADJ
asir-36063	147	21	lemma	lemma	PROPN
asir-36063	147	22	.	.	PUNCT
asir-36063	148	1	lemma	lemma	PROPN
asir-36063	148	2	3.7	3.7	NUM
asir-36063	148	3	.	.	PUNCT
asir-36063	149	1	let	let	VERB
asir-36063	149	2	𝐺	𝐺	PRON
asir-36063	149	3	be	be	AUX
asir-36063	149	4	a	a	DET
asir-36063	149	5	random	random	ADJ
asir-36063	149	6	graph	graph	NOUN
asir-36063	149	7	with	with	ADP
asir-36063	149	8	𝑛	𝑛	DET
asir-36063	149	9	node	node	NOUN
asir-36063	149	10	drawn	draw	VERB
asir-36063	149	11	accordingly	accordingly	ADV
asir-36063	149	12	to	to	ADP
asir-36063	149	13	the	the	DET
asir-36063	149	14	directed	direct	VERB
asir-36063	149	15	stochastic	stochastic	ADJ
asir-36063	149	16	block	block	NOUN
asir-36063	149	17	model	model	NOUN
asir-36063	149	18	on	on	ADP
asir-36063	149	19	two	two	NUM
asir-36063	149	20	communities	community	NOUN
asir-36063	149	21	with	with	ADP
asir-36063	149	22	in	in	ADP
asir-36063	149	23	-	-	PUNCT
asir-36063	149	24	community	community	NOUN
asir-36063	149	25	edge	edge	NOUN
asir-36063	149	26	probability	probability	NOUN
asir-36063	149	27	𝑝	𝑝	PROPN
asir-36063	149	28	and	and	CCONJ
asir-36063	149	29	cross	cross	ADJ
asir-36063	149	30	-	-	ADJ
asir-36063	149	31	community	community	ADJ
asir-36063	149	32	edge	edge	NOUN
asir-36063	149	33	probability	probability	NOUN
asir-36063	149	34	𝑞.	𝑞.	NOUN
asir-36063	149	35	let	let	VERB
asir-36063	149	36	𝑝	𝑝	NOUN
asir-36063	149	37	=	=	SYM
asir-36063	149	38	alog⁡(𝑛)/𝑛	alog⁡(𝑛)/𝑛	PROPN
asir-36063	149	39	and	and	CCONJ
asir-36063	149	40	𝑞	𝑞	X
asir-36063	149	41	=	=	SYM
asir-36063	149	42	𝑏log⁡(𝑛)/𝑛	𝑏log⁡(𝑛)/𝑛	PROPN
asir-36063	149	43	,	,	PUNCT
asir-36063	149	44	where	where	SCONJ
asir-36063	149	45	𝑎	𝑎	X
asir-36063	149	46	>	>	X
asir-36063	149	47	𝑏	𝑏	PROPN
asir-36063	149	48	are	be	AUX
asir-36063	149	49	constant	constant	ADJ
asir-36063	149	50	.	.	PUNCT
asir-36063	150	1	then	then	ADV
asir-36063	150	2	for	for	ADP
asir-36063	150	3	any	any	DET
asir-36063	150	4	constant	constant	ADJ
asir-36063	150	5	>	>	X
asir-36063	150	6	0	0	NUM
asir-36063	150	7	:	:	PUNCT
asir-36063	150	8	if	if	SCONJ
asir-36063	150	9	√𝑎	√𝑎	ADJ
asir-36063	150	10	−	−	PROPN
asir-36063	150	11	√𝑏	√𝑏	NOUN
asir-36063	150	12	>	>	X
asir-36063	150	13	√2	√2	NOUN
asir-36063	150	14	(	(	PUNCT
asir-36063	150	15	3.11	3.11	NUM
asir-36063	150	16	)	)	PUNCT
asir-36063	150	17	then	then	ADV
asir-36063	150	18	with	with	ADP
asir-36063	150	19	high	high	ADJ
asir-36063	150	20	probability	probability	NOUN
asir-36063	150	21	min	min	NOUN
asir-36063	150	22	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	150	23	]	]	PUNCT
asir-36063	150	24	 	 	SPACE
asir-36063	150	25	(	(	PUNCT
asir-36063	150	26	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	150	27	+	+	CCONJ
asir-36063	150	28	−	−	PROPN
asir-36063	150	29	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	150	30	−	−	NOUN
asir-36063	150	31	)	)	PUNCT
asir-36063	150	32	≥	≥	NOUN
asir-36063	150	33	𝚫	𝚫	PROPN
asir-36063	150	34	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	150	35	)	)	PUNCT
asir-36063	150	36	𝔼[deg𝐶	𝔼[deg𝐶	NOUN
asir-36063	150	37	+	+	NOUN
asir-36063	150	38	⁡(𝑖	⁡(𝑖	NOUN
asir-36063	150	39	)	)	PUNCT
asir-36063	151	1	−	−	ADP
asir-36063	151	2	deg𝐶	deg𝐶	ADJ
asir-36063	151	3	−⁡(𝑖	−⁡(𝑖	PROPN
asir-36063	151	4	)	)	PUNCT
asir-36063	151	5	]	]	PUNCT
asir-36063	152	1	(	(	PUNCT
asir-36063	152	2	3.12	3.12	NUM
asir-36063	152	3	)	)	PUNCT
asir-36063	152	4	4	4	NUM
asir-36063	152	5	.	.	PUNCT
asir-36063	152	6	experiments	experiment	VERB
asir-36063	152	7	the	the	DET
asir-36063	152	8	goal	goal	NOUN
asir-36063	152	9	of	of	ADP
asir-36063	152	10	the	the	DET
asir-36063	152	11	numerical	numerical	ADJ
asir-36063	152	12	experiments	experiment	NOUN
asir-36063	152	13	is	be	AUX
asir-36063	152	14	to	to	PART
asir-36063	152	15	confirm	confirm	VERB
asir-36063	152	16	our	our	PRON
asir-36063	152	17	theoretical	theoretical	ADJ
asir-36063	152	18	results	result	NOUN
asir-36063	152	19	above	above	ADV
asir-36063	152	20	.	.	PUNCT
asir-36063	153	1	let	let	VERB
asir-36063	153	2	𝑛	𝑛	PRON
asir-36063	153	3	=	=	SYM
asir-36063	153	4	100	100	NUM
asir-36063	153	5	.	.	PUNCT
asir-36063	154	1	for	for	ADP
asir-36063	154	2	each	each	DET
asir-36063	154	3	(	(	PUNCT
asir-36063	154	4	𝑎	𝑎	PROPN
asir-36063	154	5	,	,	PUNCT
asir-36063	154	6	𝑏	𝑏	NOUN
asir-36063	154	7	)	)	PUNCT
asir-36063	154	8	pair	pair	NOUN
asir-36063	154	9	we	we	PRON
asir-36063	154	10	generate	generate	VERB
asir-36063	154	11	the	the	DET
asir-36063	154	12	dsbm	dsbm	ADJ
asir-36063	154	13	25	25	NUM
asir-36063	154	14	times	time	NOUN
asir-36063	154	15	and	and	CCONJ
asir-36063	154	16	apply	apply	VERB
asir-36063	154	17	the	the	DET
asir-36063	154	18	sdp	sdp	NOUN
asir-36063	154	19	relaxation	relaxation	NOUN
asir-36063	154	20	to	to	PART
asir-36063	154	21	recover	recover	VERB
asir-36063	154	22	the	the	DET
asir-36063	154	23	community	community	NOUN
asir-36063	154	24	.	.	PUNCT
asir-36063	155	1	figure	figure	NOUN
asir-36063	155	2	1	1	NUM
asir-36063	155	3	visualizes	visualize	VERB
asir-36063	155	4	the	the	DET
asir-36063	155	5	accuracy	accuracy	NOUN
asir-36063	155	6	v.s.	v.s.	X
asir-36063	156	1	(	(	PUNCT
asir-36063	156	2	𝑎	𝑎	X
asir-36063	156	3	,	,	PUNCT
asir-36063	156	4	𝑏	𝑏	NOUN
asir-36063	156	5	)	)	PUNCT
asir-36063	156	6	.	.	PUNCT
asir-36063	157	1	the	the	DET
asir-36063	157	2	accuracy	accuracy	NOUN
asir-36063	157	3	is	be	AUX
asir-36063	157	4	calculated	calculate	VERB
asir-36063	157	5	as	as	SCONJ
asir-36063	157	6	follows	follow	VERB
asir-36063	157	7	1	1	NUM
asir-36063	157	8	2𝑛	2𝑛	NUM
asir-36063	157	9	∑	∑	PROPN
asir-36063	157	10	  	  	SPACE
asir-36063	157	11	𝑛	𝑛	DET
asir-36063	157	12	𝑖=1	𝑖=1	PROPN
asir-36063	157	13	𝟏{𝑔𝑖=𝑥𝑖	𝟏{𝑔𝑖=𝑥𝑖	PROPN
asir-36063	157	14	}	}	PUNCT
asir-36063	157	15	where	where	SCONJ
asir-36063	157	16	𝟏{⋅	𝟏{⋅	NOUN
asir-36063	157	17	}	}	PUNCT
asir-36063	157	18	is	be	AUX
asir-36063	157	19	the	the	DET
asir-36063	157	20	indicator	indicator	NOUN
asir-36063	157	21	function	function	NOUN
asir-36063	157	22	and	and	CCONJ
asir-36063	157	23	𝒈	𝒈	PROPN
asir-36063	157	24	is	be	AUX
asir-36063	157	25	the	the	DET
asir-36063	157	26	ground	ground	NOUN
asir-36063	157	27	-	-	PUNCT
asir-36063	157	28	truth	truth	NOUN
asir-36063	157	29	and	and	CCONJ
asir-36063	157	30	𝒙	𝒙	NOUN
asir-36063	157	31	is	be	AUX
asir-36063	157	32	the	the	DET
asir-36063	157	33	solution	solution	NOUN
asir-36063	157	34	of	of	ADP
asir-36063	157	35	sdp	sdp	NOUN
asir-36063	157	36	relaxation	relaxation	NOUN
asir-36063	157	37	.	.	PUNCT
asir-36063	158	1	as	as	SCONJ
asir-36063	158	2	depicted	depict	VERB
asir-36063	158	3	in	in	ADP
asir-36063	158	4	figure	figure	NOUN
asir-36063	158	5	1	1	NUM
asir-36063	158	6	,	,	PUNCT
asir-36063	158	7	the	the	DET
asir-36063	158	8	empirical	empirical	ADJ
asir-36063	158	9	boundary	boundary	NOUN
asir-36063	158	10	between	between	ADP
asir-36063	158	11	the	the	DET
asir-36063	158	12	success	success	NOUN
asir-36063	158	13	region	region	NOUN
asir-36063	158	14	and	and	CCONJ
asir-36063	158	15	the	the	DET
asir-36063	158	16	failure	failure	NOUN
asir-36063	158	17	region	region	NOUN
asir-36063	158	18	almost	almost	ADV
asir-36063	158	19	aligns	align	VERB
asir-36063	158	20	with	with	ADP
asir-36063	158	21	the	the	DET
asir-36063	158	22	curve	curve	NOUN
asir-36063	158	23	√𝑎	√𝑎	PUNCT
asir-36063	158	24	−	−	PROPN
asir-36063	158	25	√𝑏	√𝑏	NOUN
asir-36063	159	1	=	=	SYM
asir-36063	159	2	√2	√2	NOUN
asir-36063	159	3	,	,	PUNCT
asir-36063	159	4	which	which	PRON
asir-36063	159	5	suggests	suggest	VERB
asir-36063	159	6	that	that	SCONJ
asir-36063	159	7	our	our	PRON
asir-36063	159	8	proved	prove	VERB
asir-36063	159	9	information	information	NOUN
asir-36063	159	10	-	-	PUNCT
asir-36063	159	11	theoretical	theoretical	ADJ
asir-36063	159	12	threshold	threshold	NOUN
asir-36063	159	13	is	be	AUX
asir-36063	159	14	tight	tight	ADJ
asir-36063	159	15	.	.	PUNCT
asir-36063	160	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	160	2	applied	apply	VERB
asir-36063	160	3	science	science	NOUN
asir-36063	160	4	and	and	CCONJ
asir-36063	160	5	innovative	innovative	ADJ
asir-36063	160	6	research	research	NOUN
asir-36063	160	7	vol	vol	NOUN
asir-36063	160	8	.	.	PROPN
asir-36063	161	1	8	8	NUM
asir-36063	161	2	,	,	PUNCT
asir-36063	161	3	no	no	INTJ
asir-36063	161	4	.	.	NOUN
asir-36063	161	5	1	1	NUM
asir-36063	161	6	,	,	PUNCT
asir-36063	161	7	2024	2024	NUM
asir-36063	161	8	71	71	NUM
asir-36063	161	9	published	publish	VERB
asir-36063	161	10	by	by	ADP
asir-36063	161	11	scholink	scholink	PROPN
asir-36063	161	12	inc	inc	PROPN
asir-36063	161	13	.	.	PROPN
asir-36063	161	14	figure	figure	NOUN
asir-36063	161	15	1	1	NUM
asir-36063	161	16	.	.	PUNCT
asir-36063	162	1	accuracy	accuracy	NOUN
asir-36063	162	2	of	of	ADP
asir-36063	162	3	the	the	DET
asir-36063	162	4	sdp	sdp	NOUN
asir-36063	162	5	relaxation	relaxation	NOUN
asir-36063	162	6	under	under	ADP
asir-36063	162	7	different	different	ADJ
asir-36063	162	8	𝒂	𝒂	X
asir-36063	162	9	and	and	CCONJ
asir-36063	162	10	'	'	PUNCT
asir-36063	162	11	s.	s.	PROPN
asir-36063	162	12	each	each	PRON
asir-36063	162	13	(	(	PUNCT
asir-36063	162	14	𝒂	𝒂	X
asir-36063	162	15	,	,	PUNCT
asir-36063	162	16	𝒃	𝒃	NOUN
asir-36063	162	17	)	)	PUNCT
asir-36063	162	18	pair	pair	NOUN
asir-36063	162	19	is	be	AUX
asir-36063	162	20	subject	subject	ADJ
asir-36063	162	21	to	to	ADP
asir-36063	162	22	25	25	NUM
asir-36063	162	23	experiments	experiment	NOUN
asir-36063	162	24	a	a	DET
asir-36063	162	25	proof	proof	NOUN
asir-36063	162	26	for	for	ADP
asir-36063	162	27	theorem	theorem	ADJ
asir-36063	162	28	3.1	3.1	NUM
asir-36063	162	29	in	in	ADP
asir-36063	162	30	this	this	DET
asir-36063	162	31	section	section	NOUN
asir-36063	162	32	,	,	PUNCT
asir-36063	162	33	we	we	PRON
asir-36063	162	34	are	be	AUX
asir-36063	162	35	trying	try	VERB
asir-36063	162	36	to	to	PART
asir-36063	162	37	discover	discover	VERB
asir-36063	162	38	the	the	DET
asir-36063	162	39	condition	condition	NOUN
asir-36063	162	40	when	when	SCONJ
asir-36063	162	41	mle	mle	PROPN
asir-36063	162	42	can	can	AUX
asir-36063	162	43	not	not	PART
asir-36063	162	44	fully	fully	ADV
asir-36063	162	45	recover	recover	VERB
asir-36063	162	46	the	the	DET
asir-36063	162	47	communities	community	NOUN
asir-36063	162	48	in	in	ADP
asir-36063	162	49	dsbm	dsbm	ADJ
asir-36063	162	50	and	and	CCONJ
asir-36063	162	51	the	the	DET
asir-36063	162	52	condition	condition	NOUN
asir-36063	162	53	's	's	PART
asir-36063	162	54	proof	proof	NOUN
asir-36063	162	55	.	.	PUNCT
asir-36063	163	1	let	let	VERB
asir-36063	163	2	𝐺	𝐺	PRON
asir-36063	163	3	be	be	AUX
asir-36063	163	4	a	a	DET
asir-36063	163	5	graph	graph	NOUN
asir-36063	163	6	drawn	draw	VERB
asir-36063	163	7	from	from	ADP
asir-36063	163	8	𝒢(𝑛	𝒢(𝑛	NOUN
asir-36063	163	9	,	,	PUNCT
asir-36063	163	10	𝑝	𝑝	PROPN
asir-36063	163	11	,	,	PUNCT
asir-36063	163	12	𝑞	𝑞	NOUN
asir-36063	163	13	)	)	PUNCT
asir-36063	163	14	,	,	PUNCT
asir-36063	163	15	let	let	VERB
asir-36063	163	16	𝑝	𝑝	NOUN
asir-36063	163	17	=	=	PUNCT
asir-36063	163	18	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	163	19	)	)	PUNCT
asir-36063	163	20	𝑛	𝑛	NOUN
asir-36063	163	21	and	and	CCONJ
asir-36063	163	22	𝑞	𝑞	X
asir-36063	163	23	=	=	PUNCT
asir-36063	163	24	𝑏log⁡(𝑛	𝑏log⁡(𝑛	PROPN
asir-36063	163	25	)	)	PUNCT
asir-36063	163	26	𝑛	𝑛	NOUN
asir-36063	163	27	,	,	PUNCT
asir-36063	163	28	𝑎	𝑎	PROPN
asir-36063	163	29	>	>	X
asir-36063	163	30	𝑏	𝑏	NOUN
asir-36063	163	31	,	,	PUNCT
asir-36063	163	32	if	if	SCONJ
asir-36063	163	33	√𝑎	√𝑎	ADJ
asir-36063	163	34	−	−	NOUN
asir-36063	163	35	√𝑏	√𝑏	NOUN
asir-36063	163	36	<	<	X
asir-36063	163	37	√2	√2	NOUN
asir-36063	163	38	,	,	PUNCT
asir-36063	163	39	then	then	ADV
asir-36063	163	40	we	we	PRON
asir-36063	163	41	need	need	VERB
asir-36063	163	42	to	to	PART
asir-36063	163	43	prove	prove	VERB
asir-36063	163	44	that	that	SCONJ
asir-36063	163	45	for	for	ADP
asir-36063	163	46	this	this	DET
asir-36063	163	47	condition	condition	NOUN
asir-36063	163	48	the	the	DET
asir-36063	163	49	exact	exact	ADJ
asir-36063	163	50	recovery	recovery	NOUN
asir-36063	163	51	of	of	ADP
asir-36063	163	52	dsbm	dsbm	ADJ
asir-36063	163	53	is	be	AUX
asir-36063	163	54	unsolvable	unsolvable	ADJ
asir-36063	163	55	,	,	PUNCT
asir-36063	163	56	and	and	CCONJ
asir-36063	163	57	therefore	therefore	ADV
asir-36063	163	58	mle	mle	PROPN
asir-36063	163	59	fails	fail	VERB
asir-36063	163	60	.	.	PUNCT
asir-36063	164	1	we	we	PRON
asir-36063	164	2	define	define	VERB
asir-36063	164	3	mle	mle	NOUN
asir-36063	164	4	for	for	ADP
asir-36063	164	5	dsbm	dsbm	ADJ
asir-36063	164	6	to	to	PART
asir-36063	164	7	be	be	AUX
asir-36063	164	8	:	:	PUNCT
asir-36063	164	9	𝐿(𝑥	𝐿(𝑥	ADP
asir-36063	164	10	,	,	PUNCT
asir-36063	164	11	𝑦	𝑦	X
asir-36063	164	12	)	)	PUNCT
asir-36063	164	13	=	=	SYM
asir-36063	164	14	∏	∏	PROPN
asir-36063	164	15	 	 	SPACE
asir-36063	164	16	𝑖,𝑗	𝑖,𝑗	PROPN
asir-36063	164	17	𝑃𝑖,𝑗	𝑃𝑖,𝑗	PROPN
asir-36063	164	18	𝐴𝑖,𝑗(1	𝐴𝑖,𝑗(1	NOUN
asir-36063	164	19	−	−	PROPN
asir-36063	164	20	𝑃𝑖𝑗	𝑃𝑖𝑗	PROPN
asir-36063	164	21	)	)	PUNCT
asir-36063	164	22	1−𝐴𝑖,𝑗	1−𝐴𝑖,𝑗	PROPN
asir-36063	164	23	(	(	PUNCT
asir-36063	164	24	a.1	a.1	NOUN
asir-36063	164	25	)	)	PUNCT
asir-36063	164	26	definition	definition	NOUN
asir-36063	164	27	a.1	a.1	NOUN
asir-36063	164	28	.	.	PUNCT
asir-36063	165	1	we	we	PRON
asir-36063	165	2	define	define	VERB
asir-36063	165	3	a	a	DET
asir-36063	165	4	pair	pair	NOUN
asir-36063	165	5	of	of	ADP
asir-36063	165	6	bad	bad	ADJ
asir-36063	165	7	vertices	vertex	NOUN
asir-36063	165	8	in	in	ADP
asir-36063	165	9	rows	row	NOUN
asir-36063	165	10	,	,	PUNCT
asir-36063	165	11	in	in	ADP
asir-36063	165	12	columns	column	NOUN
asir-36063	165	13	,	,	PUNCT
asir-36063	165	14	or	or	CCONJ
asir-36063	165	15	in	in	ADP
asir-36063	165	16	rows	row	NOUN
asir-36063	165	17	and	and	CCONJ
asir-36063	165	18	columns	column	NOUN
asir-36063	165	19	respectively	respectively	ADV
asir-36063	165	20	by	by	ADP
asir-36063	165	21	:	:	PUNCT
asir-36063	165	22	𝐵𝑅(𝐺	𝐵𝑅(𝐺	PROPN
asir-36063	165	23	):	):	PUNCT
asir-36063	165	24	=	=	SYM
asir-36063	165	25	{	{	PUNCT
asir-36063	165	26	(	(	PUNCT
asir-36063	165	27	𝑢	𝑢	X
asir-36063	165	28	,	,	PUNCT
asir-36063	165	29	𝑣	𝑣	NOUN
asir-36063	165	30	):	):	PUNCT
asir-36063	165	31	𝑢	𝑢	PROPN
asir-36063	165	32	∈	∈	NOUN
asir-36063	165	33	𝐶1	𝐶1	PROPN
asir-36063	165	34	𝑅	𝑅	PROPN
asir-36063	165	35	,	,	PUNCT
asir-36063	165	36	𝑣	𝑣	PROPN
asir-36063	165	37	∈	∈	PROPN
asir-36063	165	38	𝐶2	𝐶2	PROPN
asir-36063	165	39	𝑅	𝑅	PROPN
asir-36063	165	40	,	,	PUNCT
asir-36063	165	41	𝐿(	𝐿(	PROPN
asir-36063	165	42	�	�	PROPN
asir-36063	165	43	̃	̃	PROPN
asir-36063	165	44	�	�	PROPN
asir-36063	165	45	,	,	PUNCT
asir-36063	165	46	𝑦	𝑦	NOUN
asir-36063	165	47	)	)	PUNCT
asir-36063	165	48	>	>	X
asir-36063	166	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	166	2	,	,	PUNCT
asir-36063	166	3	𝑦	𝑦	NOUN
asir-36063	166	4	)	)	PUNCT
asir-36063	166	5	}	}	PUNCT
asir-36063	166	6	𝐵𝐶(𝐺	𝐵𝐶(𝐺	NOUN
asir-36063	166	7	):	):	PUNCT
asir-36063	166	8	=	=	SYM
asir-36063	166	9	{	{	PUNCT
asir-36063	166	10	(	(	PUNCT
asir-36063	166	11	𝑢	𝑢	X
asir-36063	166	12	,	,	PUNCT
asir-36063	166	13	𝑣	𝑣	NOUN
asir-36063	166	14	):	):	PUNCT
asir-36063	166	15	𝑢	𝑢	PROPN
asir-36063	166	16	∈	∈	PROPN
asir-36063	166	17	𝐶1	𝐶1	PROPN
asir-36063	166	18	𝐶	𝐶	PROPN
asir-36063	166	19	,	,	PUNCT
asir-36063	166	20	𝑣	𝑣	PROPN
asir-36063	166	21	∈	∈	PROPN
asir-36063	166	22	𝐶2	𝐶2	PROPN
asir-36063	166	23	𝐶	𝐶	PROPN
asir-36063	166	24	,	,	PUNCT
asir-36063	166	25	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	166	26	,	,	PUNCT
asir-36063	166	27	�	�	PROPN
asir-36063	166	28	̃	̃	PROPN
asir-36063	166	29	�	�	PROPN
asir-36063	166	30	)	)	PUNCT
asir-36063	166	31	>	>	X
asir-36063	166	32	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	166	33	,	,	PUNCT
asir-36063	166	34	𝑦	𝑦	NOUN
asir-36063	166	35	)	)	PUNCT
asir-36063	166	36	}	}	PUNCT
asir-36063	166	37	𝐵𝑅,𝐶(𝐺	𝐵𝑅,𝐶(𝐺	NOUN
asir-36063	166	38	):	):	PUNCT
asir-36063	166	39	=	=	SYM
asir-36063	166	40	{	{	PUNCT
asir-36063	166	41	(	(	PUNCT
asir-36063	166	42	𝑢	𝑢	X
asir-36063	166	43	,	,	PUNCT
asir-36063	166	44	𝑣	𝑣	NOUN
asir-36063	166	45	):	):	PUNCT
asir-36063	166	46	𝑢	𝑢	PROPN
asir-36063	166	47	∈	∈	PROPN
asir-36063	166	48	𝐶1	𝐶1	NOUN
asir-36063	166	49	,	,	PUNCT
asir-36063	166	50	𝑣	𝑣	PROPN
asir-36063	166	51	∈	∈	PROPN
asir-36063	166	52	𝐶2	𝐶2	PROPN
asir-36063	166	53	,	,	PUNCT
asir-36063	166	54	𝐿(	𝐿(	PROPN
asir-36063	166	55	�	�	PROPN
asir-36063	166	56	̃	̃	PROPN
asir-36063	166	57	�	�	PROPN
asir-36063	166	58	,	,	PUNCT
asir-36063	166	59	�	�	PROPN
asir-36063	166	60	̃	̃	PROPN
asir-36063	166	61	�	�	PROPN
asir-36063	166	62	)	)	PUNCT
asir-36063	166	63	>	>	X
asir-36063	166	64	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	166	65	,	,	PUNCT
asir-36063	166	66	𝑦	𝑦	NOUN
asir-36063	166	67	)	)	PUNCT
asir-36063	166	68	}	}	PUNCT
asir-36063	166	69	(	(	PUNCT
asir-36063	166	70	a.2	a.2	X
asir-36063	166	71	)	)	PUNCT
asir-36063	166	72	then	then	ADV
asir-36063	166	73	we	we	PRON
asir-36063	166	74	are	be	AUX
asir-36063	166	75	trying	try	VERB
asir-36063	166	76	to	to	PART
asir-36063	166	77	prove	prove	VERB
asir-36063	166	78	that	that	SCONJ
asir-36063	166	79	for	for	ADP
asir-36063	166	80	√𝑎	√𝑎	ADJ
asir-36063	166	81	−	−	PROPN
asir-36063	166	82	√𝑏	√𝑏	NOUN
asir-36063	166	83	<	<	X
asir-36063	166	84	√2	√2	NOUN
asir-36063	166	85	,	,	PUNCT
asir-36063	166	86	there	there	PRON
asir-36063	166	87	exists	exist	VERB
asir-36063	166	88	at	at	ADP
asir-36063	166	89	least	least	ADV
asir-36063	166	90	one	one	NUM
asir-36063	166	91	bad	bad	ADJ
asir-36063	166	92	vertex	vertex	NOUN
asir-36063	166	93	in	in	ADP
asir-36063	166	94	rows	row	NOUN
asir-36063	166	95	or	or	CCONJ
asir-36063	166	96	columns	column	NOUN
asir-36063	166	97	,	,	PUNCT
asir-36063	166	98	which	which	PRON
asir-36063	166	99	would	would	AUX
asir-36063	166	100	result	result	VERB
asir-36063	166	101	in	in	ADP
asir-36063	166	102	max{𝐿(	max{𝐿(	PROPN
asir-36063	166	103	�	�	PROPN
asir-36063	166	104	̃	̃	PROPN
asir-36063	166	105	�	�	PROPN
asir-36063	166	106	,	,	PUNCT
asir-36063	166	107	𝑦	𝑦	NOUN
asir-36063	166	108	)	)	PUNCT
asir-36063	166	109	,	,	PUNCT
asir-36063	166	110	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	166	111	,	,	PUNCT
asir-36063	166	112	�	�	PROPN
asir-36063	166	113	̃	̃	PROPN
asir-36063	166	114	�	�	PROPN
asir-36063	166	115	)	)	PUNCT
asir-36063	166	116	,	,	PUNCT
asir-36063	166	117	𝐿(	𝐿(	PROPN
asir-36063	166	118	�	�	PROPN
asir-36063	166	119	̃	̃	PROPN
asir-36063	166	120	�	�	PROPN
asir-36063	166	121	,	,	PUNCT
asir-36063	166	122	�	�	PROPN
asir-36063	166	123	̃	̃	PROPN
asir-36063	166	124	�	�	PROPN
asir-36063	166	125	)	)	PUNCT
asir-36063	166	126	≥	≥	NOUN
asir-36063	166	127	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	166	128	,	,	PUNCT
asir-36063	166	129	𝑦	𝑦	NOUN
asir-36063	166	130	)	)	PUNCT
asir-36063	166	131	}	}	PUNCT
asir-36063	166	132	.	.	PUNCT
asir-36063	167	1	we	we	PRON
asir-36063	167	2	define	define	VERB
asir-36063	167	3	a	a	DET
asir-36063	167	4	pair	pair	NOUN
asir-36063	167	5	of	of	ADP
asir-36063	167	6	bad	bad	ADJ
asir-36063	167	7	vertex	vertex	NOUN
asir-36063	167	8	(	(	PUNCT
asir-36063	167	9	𝑢	𝑢	X
asir-36063	167	10	,	,	PUNCT
asir-36063	167	11	𝑣	𝑣	NOUN
asir-36063	167	12	)	)	PUNCT
asir-36063	167	13	,	,	PUNCT
asir-36063	167	14	the	the	DET
asir-36063	167	15	relationship	relationship	NOUN
asir-36063	167	16	between	between	ADP
asir-36063	167	17	degrees	degree	NOUN
asir-36063	167	18	can	can	AUX
asir-36063	167	19	be	be	AUX
asir-36063	167	20	inferred	infer	VERB
asir-36063	167	21	from	from	ADP
asir-36063	167	22	the	the	DET
asir-36063	167	23	following	follow	VERB
asir-36063	167	24	relationship	relationship	NOUN
asir-36063	167	25	between	between	ADP
asir-36063	167	26	mle	mle	PROPN
asir-36063	167	27	:	:	PUNCT
asir-36063	167	28	𝐿(	𝐿(	PROPN
asir-36063	167	29	�	�	PROPN
asir-36063	167	30	̃	̃	PROPN
asir-36063	167	31	�	�	PROPN
asir-36063	167	32	,	,	PUNCT
asir-36063	167	33	𝑦	𝑦	NOUN
asir-36063	167	34	)	)	PUNCT
asir-36063	167	35	>	>	X
asir-36063	168	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	168	2	,	,	PUNCT
asir-36063	168	3	𝑦	𝑦	NOUN
asir-36063	168	4	)	)	PUNCT
asir-36063	168	5	→	→	SYM
asir-36063	168	6	𝑑−	𝑑−	NOUN
asir-36063	168	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	168	8	)	)	PUNCT
asir-36063	169	1	+	+	NUM
asir-36063	169	2	𝑑−	𝑑−	NOUN
asir-36063	169	3	𝑅(𝑣	𝑅(𝑣	PUNCT
asir-36063	169	4	)	)	PUNCT
asir-36063	169	5	>	>	PUNCT
asir-36063	169	6	𝑑+	𝑑+	X
asir-36063	169	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	169	8	)	)	PUNCT
asir-36063	169	9	+	+	CCONJ
asir-36063	169	10	𝑑+	𝑑+	NOUN
asir-36063	169	11	𝑅(𝑣	𝑅(𝑣	NUM
asir-36063	169	12	)	)	PUNCT
asir-36063	169	13	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	169	14	,	,	PUNCT
asir-36063	169	15	�	�	PROPN
asir-36063	169	16	̃	̃	PROPN
asir-36063	169	17	�	�	PROPN
asir-36063	169	18	)	)	PUNCT
asir-36063	169	19	>	>	X
asir-36063	170	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	170	2	,	,	PUNCT
asir-36063	170	3	𝑦	𝑦	NOUN
asir-36063	170	4	)	)	PUNCT
asir-36063	170	5	→	→	SYM
asir-36063	170	6	𝑑−	𝑑−	NOUN
asir-36063	170	7	𝐶(𝑢	𝐶(𝑢	CCONJ
asir-36063	170	8	)	)	PUNCT
asir-36063	170	9	+	+	CCONJ
asir-36063	170	10	𝑑−	𝑑−	NOUN
asir-36063	170	11	𝐶(𝑣	𝐶(𝑣	NOUN
asir-36063	170	12	)	)	PUNCT
asir-36063	170	13	>	>	X
asir-36063	170	14	𝑑+	𝑑+	X
asir-36063	170	15	𝐶(𝑢	𝐶(𝑢	X
asir-36063	170	16	)	)	PUNCT
asir-36063	171	1	+	+	CCONJ
asir-36063	171	2	𝑑+	𝑑+	X
asir-36063	171	3	𝐶(𝑣)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(a	𝐶(𝑣)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(a	NOUN
asir-36063	171	4	.	.	PUNCT
asir-36063	172	1	3	3	X
asir-36063	172	2	)	)	PUNCT
asir-36063	172	3	𝐿(	𝐿(	PROPN
asir-36063	172	4	�	�	PROPN
asir-36063	172	5	̃	̃	PROPN
asir-36063	172	6	�	�	PROPN
asir-36063	172	7	,	,	PUNCT
asir-36063	172	8	�	�	PROPN
asir-36063	172	9	̃	̃	PROPN
asir-36063	172	10	�	�	PROPN
asir-36063	172	11	)	)	PUNCT
asir-36063	172	12	>	>	X
asir-36063	173	1	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	173	2	,	,	PUNCT
asir-36063	173	3	𝑦	𝑦	NOUN
asir-36063	173	4	)	)	PUNCT
asir-36063	173	5	→	→	SYM
asir-36063	173	6	𝑑−	𝑑−	NOUN
asir-36063	173	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	173	8	)	)	PUNCT
asir-36063	174	1	+	+	NUM
asir-36063	174	2	𝑑−	𝑑−	NOUN
asir-36063	174	3	𝑅(𝑣	𝑅(𝑣	PUNCT
asir-36063	174	4	)	)	PUNCT
asir-36063	175	1	+	+	NUM
asir-36063	175	2	𝑑−	𝑑−	NOUN
asir-36063	175	3	𝐶(𝑢	𝐶(𝑢	CCONJ
asir-36063	175	4	)	)	PUNCT
asir-36063	176	1	+	+	CCONJ
asir-36063	176	2	𝑑−	𝑑−	NOUN
asir-36063	176	3	𝐶(𝑣	𝐶(𝑣	NOUN
asir-36063	176	4	)	)	PUNCT
asir-36063	176	5	>	>	X
asir-36063	176	6	𝑑+	𝑑+	X
asir-36063	176	7	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	176	8	)	)	PUNCT
asir-36063	176	9	+	+	CCONJ
asir-36063	176	10	𝑑+	𝑑+	NOUN
asir-36063	176	11	𝑅(𝑣	𝑅(𝑣	NUM
asir-36063	176	12	)	)	PUNCT
asir-36063	176	13	+	+	NUM
asir-36063	176	14	𝑑+	𝑑+	X
asir-36063	176	15	𝐶(𝑢	𝐶(𝑢	NOUN
asir-36063	176	16	)	)	PUNCT
asir-36063	176	17	+	+	CCONJ
asir-36063	176	18	𝑑+	𝑑+	DET
asir-36063	176	19	𝐶(𝑣	𝐶(𝑣	NOUN
asir-36063	176	20	)	)	PUNCT
asir-36063	176	21	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	176	22	applied	apply	VERB
asir-36063	176	23	science	science	NOUN
asir-36063	176	24	and	and	CCONJ
asir-36063	176	25	innovative	innovative	ADJ
asir-36063	176	26	research	research	NOUN
asir-36063	176	27	vol	vol	NOUN
asir-36063	176	28	.	.	PROPN
asir-36063	177	1	8	8	NUM
asir-36063	177	2	,	,	PUNCT
asir-36063	177	3	no	no	INTJ
asir-36063	177	4	.	.	NOUN
asir-36063	177	5	1	1	NUM
asir-36063	177	6	,	,	PUNCT
asir-36063	177	7	2024	2024	NUM
asir-36063	177	8	72	72	NUM
asir-36063	177	9	published	publish	VERB
asir-36063	177	10	by	by	ADP
asir-36063	177	11	scholink	scholink	PROPN
asir-36063	177	12	inc	inc	PROPN
asir-36063	177	13	.	.	PROPN
asir-36063	178	1	since	since	SCONJ
asir-36063	178	2	if	if	SCONJ
asir-36063	178	3	𝐿(	𝐿(	PROPN
asir-36063	178	4	�	�	PROPN
asir-36063	178	5	̃	̃	PROPN
asir-36063	178	6	�	�	PROPN
asir-36063	178	7	,	,	PUNCT
asir-36063	178	8	𝑦	𝑦	NOUN
asir-36063	178	9	)	)	PUNCT
asir-36063	178	10	>	>	X
asir-36063	178	11	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	178	12	,	,	PUNCT
asir-36063	178	13	𝑦	𝑦	NOUN
asir-36063	178	14	)	)	PUNCT
asir-36063	178	15	,	,	PUNCT
asir-36063	178	16	then	then	ADV
asir-36063	178	17	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	178	18	,	,	PUNCT
asir-36063	178	19	�	�	PROPN
asir-36063	178	20	̃	̃	PROPN
asir-36063	178	21	�	�	PROPN
asir-36063	178	22	)	)	PUNCT
asir-36063	178	23	>	>	X
asir-36063	178	24	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	178	25	,	,	PUNCT
asir-36063	178	26	𝑦)or⁡𝐿(	𝑦)or⁡𝐿(	NOUN
asir-36063	178	27	�	�	PROPN
asir-36063	178	28	̃	̃	PROPN
asir-36063	178	29	�	�	PROPN
asir-36063	178	30	,	,	PUNCT
asir-36063	178	31	�	�	PROPN
asir-36063	178	32	̃	̃	PROPN
asir-36063	178	33	�	�	PROPN
asir-36063	178	34	)	)	PUNCT
asir-36063	178	35	>	>	X
asir-36063	178	36	𝐿(𝑥	𝐿(𝑥	PROPN
asir-36063	178	37	,	,	PUNCT
asir-36063	178	38	𝑦	𝑦	NOUN
asir-36063	178	39	)	)	PUNCT
asir-36063	178	40	.	.	PUNCT
asir-36063	179	1	hence	hence	ADV
asir-36063	179	2	,	,	PUNCT
asir-36063	179	3	it	it	PRON
asir-36063	179	4	is	be	AUX
asir-36063	179	5	enough	enough	ADJ
asir-36063	179	6	to	to	PART
asir-36063	179	7	only	only	ADV
asir-36063	179	8	study	study	VERB
asir-36063	179	9	the	the	DET
asir-36063	179	10	bad	bad	ADJ
asir-36063	179	11	vertices	vertex	NOUN
asir-36063	179	12	in	in	ADP
asir-36063	179	13	rows	row	NOUN
asir-36063	179	14	or	or	CCONJ
asir-36063	179	15	in	in	ADP
asir-36063	179	16	columns	column	NOUN
asir-36063	179	17	.	.	PUNCT
asir-36063	180	1	definition	definition	NOUN
asir-36063	180	2	a.2	a.2	PUNCT
asir-36063	180	3	.	.	PUNCT
asir-36063	181	1	we	we	PRON
asir-36063	181	2	define	define	VERB
asir-36063	181	3	a	a	DET
asir-36063	181	4	set	set	NOUN
asir-36063	181	5	of	of	ADP
asir-36063	181	6	bad	bad	ADJ
asir-36063	181	7	vertices	vertex	NOUN
asir-36063	181	8	in	in	ADP
asir-36063	181	9	rows	row	NOUN
asir-36063	181	10	:	:	PUNCT
asir-36063	181	11	𝐵𝑖	𝐵𝑖	PROPN
asir-36063	181	12	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	181	13	)	)	PUNCT
asir-36063	181	14	=	=	PRON
asir-36063	181	15	{	{	PUNCT
asir-36063	181	16	∃𝑢	∃𝑢	PROPN
asir-36063	181	17	:	:	PUNCT
asir-36063	181	18	𝑢	𝑢	PROPN
asir-36063	181	19	∈	∈	PROPN
asir-36063	181	20	𝐶𝑖	𝐶𝑖	PROPN
asir-36063	181	21	𝑅	𝑅	PROPN
asir-36063	181	22	,	,	PUNCT
asir-36063	181	23	𝑑+	𝑑+	X
asir-36063	181	24	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	181	25	)	)	PUNCT
asir-36063	181	26	≤	≤	NUM
asir-36063	181	27	𝑑−	𝑑−	ADV
asir-36063	181	28	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	181	29	)	)	PUNCT
asir-36063	181	30	−	−	PROPN
asir-36063	181	31	1	1	NUM
asir-36063	181	32	}	}	PUNCT
asir-36063	181	33	,	,	PUNCT
asir-36063	181	34	𝑖	𝑖	SYM
asir-36063	181	35	=	=	SYM
asir-36063	181	36	1,2	1,2	NUM
asir-36063	181	37	(	(	PUNCT
asir-36063	181	38	a.4	a.4	NOUN
asir-36063	181	39	)	)	PUNCT
asir-36063	181	40	where	where	SCONJ
asir-36063	181	41	𝑖	𝑖	PRON
asir-36063	181	42	represents	represent	VERB
asir-36063	181	43	the	the	DET
asir-36063	181	44	community	community	NOUN
asir-36063	181	45	.	.	PUNCT
asir-36063	182	1	lemma	lemma	PROPN
asir-36063	182	2	a.1	a.1	PROPN
asir-36063	182	3	.	.	PUNCT
asir-36063	183	1	if	if	SCONJ
asir-36063	183	2	𝐵1	𝐵1	PROPN
asir-36063	183	3	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	183	4	)	)	PUNCT
asir-36063	183	5	is	be	AUX
asir-36063	183	6	non	non	ADJ
asir-36063	183	7	-	-	ADJ
asir-36063	183	8	empty	empty	ADJ
asir-36063	183	9	and	and	CCONJ
asir-36063	183	10	with	with	ADP
asir-36063	183	11	high	high	ADJ
asir-36063	183	12	probability	probability	NOUN
asir-36063	183	13	,	,	PUNCT
asir-36063	183	14	then	then	ADV
asir-36063	183	15	𝐵𝑅(𝐺	𝐵𝑅(𝐺	PROPN
asir-36063	183	16	)	)	PUNCT
asir-36063	183	17	is	be	AUX
asir-36063	183	18	non	non	ADJ
asir-36063	183	19	-	-	ADJ
asir-36063	183	20	empty	empty	ADJ
asir-36063	183	21	and	and	CCONJ
asir-36063	183	22	with	with	ADP
asir-36063	183	23	non	non	ADJ
asir-36063	183	24	-	-	ADJ
asir-36063	183	25	vanishing	vanishing	ADJ
asir-36063	183	26	probability	probability	NOUN
asir-36063	183	27	.	.	PUNCT
asir-36063	184	1	proof	proof	NOUN
asir-36063	184	2	.	.	PUNCT
asir-36063	185	1	if	if	SCONJ
asir-36063	185	2	𝑢	𝑢	X
asir-36063	185	3	∈	∈	VERB
asir-36063	185	4	𝐶1	𝐶1	PROPN
asir-36063	185	5	𝑅	𝑅	NOUN
asir-36063	185	6	and	and	CCONJ
asir-36063	185	7	𝑣	𝑣	ADP
asir-36063	185	8	∈	∈	PROPN
asir-36063	185	9	𝐶2	𝐶2	PROPN
asir-36063	185	10	𝑅	𝑅	PROPN
asir-36063	185	11	such	such	ADJ
asir-36063	185	12	that	that	SCONJ
asir-36063	185	13	𝑑+	𝑑+	NOUN
asir-36063	185	14	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	185	15	)	)	PUNCT
asir-36063	185	16	≤	≤	NUM
asir-36063	185	17	𝑑−	𝑑−	ADV
asir-36063	185	18	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	185	19	)	)	PUNCT
asir-36063	185	20	−	−	PROPN
asir-36063	185	21	1	1	NUM
asir-36063	185	22	and	and	CCONJ
asir-36063	185	23	𝑑+	𝑑+	NOUN
asir-36063	185	24	𝑅(𝑣	𝑅(𝑣	NUM
asir-36063	185	25	)	)	PUNCT
asir-36063	185	26	≤	≤	NOUN
asir-36063	185	27	𝑑−	𝑑−	ADV
asir-36063	185	28	𝑅(𝑣	𝑅(𝑣	PUNCT
asir-36063	185	29	)	)	PUNCT
asir-36063	185	30	−	−	PROPN
asir-36063	185	31	1	1	NUM
asir-36063	185	32	,	,	PUNCT
asir-36063	185	33	then	then	ADV
asir-36063	185	34	combining	combine	VERB
asir-36063	185	35	these	these	PRON
asir-36063	185	36	we	we	PRON
asir-36063	185	37	get	get	VERB
asir-36063	185	38	𝑑−	𝑑−	PRON
asir-36063	185	39	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	185	40	)	)	PUNCT
asir-36063	185	41	+	+	NUM
asir-36063	185	42	𝑑−	𝑑−	NOUN
asir-36063	185	43	𝑅(𝑣	𝑅(𝑣	PUNCT
asir-36063	185	44	)	)	PUNCT
asir-36063	185	45	>	>	PUNCT
asir-36063	185	46	𝑑+	𝑑+	X
asir-36063	185	47	𝑅(𝑢	𝑅(𝑢	NUM
asir-36063	185	48	)	)	PUNCT
asir-36063	186	1	+	+	NUM
asir-36063	186	2	𝑑+	𝑑+	NOUN
asir-36063	186	3	𝑅(𝑣	𝑅(𝑣	NUM
asir-36063	186	4	)	)	PUNCT
asir-36063	186	5	.	.	PUNCT
asir-36063	187	1	then	then	ADV
asir-36063	187	2	we	we	PRON
asir-36063	187	3	have	have	VERB
asir-36063	187	4	:	:	PUNCT
asir-36063	187	5	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	187	6	∈	∈	PROPN
asir-36063	187	7	𝐵𝑅	𝐵𝑅	PROPN
asir-36063	187	8	or	or	CCONJ
asir-36063	187	9	∃𝑣	∃𝑣	NOUN
asir-36063	187	10	∈	∈	NOUN
asir-36063	187	11	𝐵𝑅	𝐵𝑅	PROPN
asir-36063	187	12	)	)	PUNCT
asir-36063	187	13	=	=	SYM
asir-36063	187	14	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	187	15	∈	∈	PROPN
asir-36063	187	16	𝐵1	𝐵1	NOUN
asir-36063	187	17	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	187	18	)	)	PUNCT
asir-36063	187	19	)	)	PUNCT
asir-36063	188	1	+	+	CCONJ
asir-36063	188	2	ℙ(∃𝑣	ℙ(∃𝑣	PROPN
asir-36063	188	3	∈	∈	PROPN
asir-36063	188	4	𝐵1	𝐵1	NOUN
asir-36063	188	5	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	188	6	)	)	PUNCT
asir-36063	188	7	)	)	PUNCT
asir-36063	189	1	−	−	PROPN
asir-36063	189	2	ℙ(∃(𝑢	ℙ(∃(𝑢	NOUN
asir-36063	189	3	,	,	PUNCT
asir-36063	189	4	𝑣	𝑣	NOUN
asir-36063	189	5	)	)	PUNCT
asir-36063	189	6	∈	∈	PROPN
asir-36063	189	7	𝐵𝑅(𝐺	𝐵𝑅(𝐺	NOUN
asir-36063	189	8	)	)	PUNCT
asir-36063	189	9	)	)	PUNCT
asir-36063	190	1	ℙ(∃(𝑢	ℙ(∃(𝑢	PROPN
asir-36063	190	2	,	,	PUNCT
asir-36063	190	3	𝑣	𝑣	NOUN
asir-36063	190	4	)	)	PUNCT
asir-36063	190	5	∈	∈	PROPN
asir-36063	190	6	𝐵𝑅(𝐺	𝐵𝑅(𝐺	NOUN
asir-36063	190	7	)	)	PUNCT
asir-36063	190	8	)	)	PUNCT
asir-36063	191	1	=	=	PUNCT
asir-36063	191	2	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	191	3	∈	∈	PROPN
asir-36063	191	4	𝐵1	𝐵1	NOUN
asir-36063	191	5	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	191	6	)	)	PUNCT
asir-36063	191	7	)	)	PUNCT
asir-36063	192	1	+	+	CCONJ
asir-36063	192	2	ℙ(∃𝑣	ℙ(∃𝑣	PROPN
asir-36063	192	3	∈	∈	PROPN
asir-36063	192	4	𝐵1	𝐵1	NOUN
asir-36063	192	5	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	192	6	)	)	PUNCT
asir-36063	192	7	)	)	PUNCT
asir-36063	193	1	−	−	PROPN
asir-36063	193	2	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	193	3	∈	∈	PROPN
asir-36063	193	4	𝐵𝑅	𝐵𝑅	PROPN
asir-36063	193	5	or	or	CCONJ
asir-36063	193	6	∃𝑣	∃𝑣	NOUN
asir-36063	193	7	∈	∈	PROPN
asir-36063	193	8	𝐵𝑅	𝐵𝑅	PROPN
asir-36063	193	9	)	)	PUNCT
asir-36063	193	10	(	(	PUNCT
asir-36063	193	11	a.5	a.5	NOUN
asir-36063	193	12	)	)	PUNCT
asir-36063	193	13	because	because	SCONJ
asir-36063	193	14	the	the	DET
asir-36063	193	15	possibility	possibility	NOUN
asir-36063	193	16	of	of	ADP
asir-36063	193	17	node	node	NOUN
asir-36063	193	18	𝑢	𝑢	PROPN
asir-36063	193	19	is	be	AUX
asir-36063	193	20	a	a	DET
asir-36063	193	21	bad	bad	ADJ
asir-36063	193	22	vertex	vertex	NOUN
asir-36063	193	23	is	be	AUX
asir-36063	193	24	the	the	DET
asir-36063	193	25	same	same	ADJ
asir-36063	193	26	as	as	SCONJ
asir-36063	193	27	node	node	ADJ
asir-36063	193	28	𝑣	𝑣	PROPN
asir-36063	193	29	is	be	AUX
asir-36063	193	30	a	a	DET
asir-36063	193	31	bad	bad	ADJ
asir-36063	193	32	vertex	vertex	NOUN
asir-36063	193	33	,	,	PUNCT
asir-36063	193	34	so	so	SCONJ
asir-36063	193	35	we	we	PRON
asir-36063	193	36	can	can	AUX
asir-36063	193	37	write	write	VERB
asir-36063	193	38	:	:	PUNCT
asir-36063	193	39	ℙ(∃(𝑢	ℙ(∃(𝑢	NOUN
asir-36063	193	40	,	,	PUNCT
asir-36063	193	41	𝑣	𝑣	NOUN
asir-36063	193	42	)	)	PUNCT
asir-36063	193	43	∈	∈	PROPN
asir-36063	193	44	𝐵𝑅(𝐺	𝐵𝑅(𝐺	NOUN
asir-36063	193	45	)	)	PUNCT
asir-36063	193	46	)	)	PUNCT
asir-36063	193	47	≤	≤	NUM
asir-36063	194	1	2ℙ(∃𝑢	2ℙ(∃𝑢	NUM
asir-36063	194	2	∈	∈	PROPN
asir-36063	194	3	𝐵1	𝐵1	NOUN
asir-36063	194	4	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	194	5	)	)	PUNCT
asir-36063	194	6	)	)	PUNCT
asir-36063	195	1	−	−	PROPN
asir-36063	195	2	1	1	NUM
asir-36063	195	3	(	(	PUNCT
asir-36063	195	4	a.6	a.6	X
asir-36063	195	5	)	)	PUNCT
asir-36063	195	6	lemma	lemma	PROPN
asir-36063	195	7	a.2	a.2	PUNCT
asir-36063	195	8	.	.	PUNCT
asir-36063	196	1	let	let	VERB
asir-36063	196	2	𝐺	𝐺	PRON
asir-36063	196	3	be	be	AUX
asir-36063	196	4	a	a	DET
asir-36063	196	5	graph	graph	NOUN
asir-36063	196	6	drawn	draw	VERB
asir-36063	196	7	from	from	ADP
asir-36063	196	8	𝒢(𝑛	𝒢(𝑛	NOUN
asir-36063	196	9	,	,	PUNCT
asir-36063	196	10	𝑝	𝑝	PROPN
asir-36063	196	11	,	,	PUNCT
asir-36063	196	12	𝑞	𝑞	NOUN
asir-36063	196	13	)	)	PUNCT
asir-36063	196	14	,	,	PUNCT
asir-36063	196	15	let	let	VERB
asir-36063	196	16	𝑝	𝑝	NOUN
asir-36063	196	17	=	=	PUNCT
asir-36063	196	18	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	196	19	)	)	PUNCT
asir-36063	196	20	𝑛	𝑛	NOUN
asir-36063	196	21	and	and	CCONJ
asir-36063	196	22	𝑞	𝑞	X
asir-36063	196	23	=	=	PUNCT
asir-36063	196	24	𝑏log⁡(𝑛	𝑏log⁡(𝑛	PROPN
asir-36063	196	25	)	)	PUNCT
asir-36063	196	26	𝑛	𝑛	NOUN
asir-36063	196	27	,	,	PUNCT
asir-36063	196	28	𝑎	𝑎	PROPN
asir-36063	196	29	>	>	X
asir-36063	196	30	𝑏	𝑏	NOUN
asir-36063	196	31	,	,	PUNCT
asir-36063	196	32	if	if	SCONJ
asir-36063	196	33	√𝑎	√𝑎	ADJ
asir-36063	196	34	−	−	NOUN
asir-36063	196	35	√𝑏	√𝑏	NOUN
asir-36063	196	36	<	<	X
asir-36063	196	37	√2	√2	NOUN
asir-36063	196	38	,	,	PUNCT
asir-36063	196	39	then	then	ADV
asir-36063	196	40	:	:	PUNCT
asir-36063	196	41	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	196	42	∈	∈	PROPN
asir-36063	196	43	𝐵1	𝐵1	NOUN
asir-36063	196	44	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	196	45	)	)	PUNCT
asir-36063	196	46	)	)	PUNCT
asir-36063	197	1	=	=	SYM
asir-36063	197	2	1	1	NUM
asir-36063	197	3	−	−	NOUN
asir-36063	197	4	𝑜(1	𝑜(1	PROPN
asir-36063	197	5	)	)	PUNCT
asir-36063	197	6	(	(	PUNCT
asir-36063	197	7	a.7	a.7	NOUN
asir-36063	197	8	)	)	PUNCT
asir-36063	197	9	proof	proof	NOUN
asir-36063	197	10	.	.	PUNCT
asir-36063	198	1	note	note	VERB
asir-36063	198	2	that	that	SCONJ
asir-36063	198	3	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	198	4	∈	∈	PROPN
asir-36063	198	5	𝐵1	𝐵1	NOUN
asir-36063	198	6	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	198	7	)	)	PUNCT
asir-36063	198	8	)	)	PUNCT
asir-36063	198	9	can	can	AUX
asir-36063	198	10	be	be	AUX
asir-36063	198	11	written	write	VERB
asir-36063	198	12	as	as	ADP
asir-36063	198	13	:	:	PUNCT
asir-36063	198	14	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	198	15	∈	∈	PROPN
asir-36063	198	16	𝐵1	𝐵1	NOUN
asir-36063	198	17	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	198	18	)	)	PUNCT
asir-36063	198	19	)	)	PUNCT
asir-36063	199	1	=	=	PUNCT
asir-36063	199	2	𝑛ℙ(𝑑−	𝑛ℙ(𝑑−	NOUN
asir-36063	199	3	𝑅	𝑅	PROPN
asir-36063	199	4	>	>	X
asir-36063	199	5	𝑑+	𝑑+	PROPN
asir-36063	199	6	𝑅	𝑅	PROPN
asir-36063	199	7	)	)	PUNCT
asir-36063	199	8	=	=	SYM
asir-36063	199	9	𝑛ℙ(bin⁡	𝑛ℙ(bin⁡	PROPN
asir-36063	199	10	(	(	PUNCT
asir-36063	199	11	𝑛	𝑛	PROPN
asir-36063	199	12	2	2	NUM
asir-36063	199	13	,	,	PUNCT
asir-36063	199	14	𝑞	𝑞	PROPN
asir-36063	199	15	)	)	PUNCT
asir-36063	199	16	>	>	X
asir-36063	199	17	bin⁡	bin⁡	PROPN
asir-36063	199	18	(	(	PUNCT
asir-36063	199	19	𝑛	𝑛	PROPN
asir-36063	199	20	2	2	NUM
asir-36063	199	21	,	,	PUNCT
asir-36063	199	22	𝑝	𝑝	NOUN
asir-36063	199	23	)	)	PUNCT
asir-36063	199	24	)	)	PUNCT
asir-36063	199	25	(	(	PUNCT
asir-36063	199	26	a.8	a.8	NOUN
asir-36063	199	27	)	)	PUNCT
asir-36063	199	28	then	then	ADV
asir-36063	199	29	,	,	PUNCT
asir-36063	199	30	we	we	PRON
asir-36063	199	31	introduce	introduce	VERB
asir-36063	199	32	a	a	DET
asir-36063	199	33	new	new	ADJ
asir-36063	199	34	definition	definition	NOUN
asir-36063	199	35	.	.	PUNCT
asir-36063	200	1	definition	definition	NOUN
asir-36063	200	2	a.3	a.3	VERB
asir-36063	200	3	.	.	PUNCT
asir-36063	201	1	let	let	VERB
asir-36063	201	2	𝑚	𝑚	PART
asir-36063	201	3	be	be	AUX
asir-36063	201	4	a	a	DET
asir-36063	201	5	natural	natural	ADJ
asir-36063	201	6	number	number	NOUN
asir-36063	201	7	,	,	PUNCT
asir-36063	201	8	𝑝	𝑝	NOUN
asir-36063	201	9	,	,	PUNCT
asir-36063	201	10	𝑞	𝑞	PROPN
asir-36063	201	11	∈	∈	PROPN
asir-36063	202	1	[	[	X
asir-36063	202	2	0,1	0,1	NUM
asir-36063	202	3	]	]	PUNCT
asir-36063	202	4	,	,	PUNCT
asir-36063	202	5	and	and	CCONJ
asir-36063	202	6	𝛿	𝛿	DET
asir-36063	202	7	∈	∈	PROPN
asir-36063	202	8	ℝ	ℝ	PROPN
asir-36063	202	9	,	,	PUNCT
asir-36063	202	10	we	we	PRON
asir-36063	202	11	define	define	VERB
asir-36063	202	12	𝑇(𝑚	𝑇(𝑚	PROPN
asir-36063	202	13	,	,	PUNCT
asir-36063	202	14	𝑝	𝑝	PROPN
asir-36063	202	15	,	,	PUNCT
asir-36063	202	16	𝑞	𝑞	NOUN
asir-36063	202	17	,	,	PUNCT
asir-36063	202	18	𝛿	𝛿	ADJ
asir-36063	202	19	)	)	PUNCT
asir-36063	202	20	=	=	SYM
asir-36063	202	21	ℙ[∑	ℙ[∑	NOUN
asir-36063	202	22	 	 	SPACE
asir-36063	202	23	𝑚	𝑚	PROPN
asir-36063	202	24	𝑖=1	𝑖=1	VERB
asir-36063	202	25	  	  	SPACE
asir-36063	202	26	(	(	PUNCT
asir-36063	203	1	𝑍𝑖	𝑍𝑖	VERB
asir-36063	203	2	−𝑊𝑖	−𝑊𝑖	VERB
asir-36063	203	3	≥	≥	NOUN
asir-36063	203	4	𝛿	𝛿	NOUN
asir-36063	203	5	)	)	PUNCT
asir-36063	203	6	]	]	PUNCT
asir-36063	203	7	(	(	PUNCT
asir-36063	203	8	a.9	a.9	PROPN
asir-36063	203	9	)	)	PUNCT
asir-36063	203	10	where	where	SCONJ
asir-36063	203	11	𝑊1	𝑊1	PROPN
asir-36063	203	12	,	,	PUNCT
asir-36063	203	13	…	…	PUNCT
asir-36063	203	14	,	,	PUNCT
asir-36063	203	15	𝑊𝑚	𝑊𝑚	PROPN
asir-36063	203	16	are	be	AUX
asir-36063	203	17	i.i.d	i.i.d	ADP
asir-36063	203	18	.	.	PUNCT
asir-36063	204	1	bernoulli(p	bernoulli(p	NOUN
asir-36063	204	2	)	)	PUNCT
asir-36063	204	3	and	and	CCONJ
asir-36063	204	4	𝑍1	𝑍1	PROPN
asir-36063	204	5	,	,	PUNCT
asir-36063	204	6	…	…	PUNCT
asir-36063	204	7	,	,	PUNCT
asir-36063	205	1	𝑍𝑚	𝑍𝑚	PROPN
asir-36063	205	2	are	be	AUX
asir-36063	205	3	i.i.d	i.i.d	ADP
asir-36063	205	4	.	.	PUNCT
asir-36063	206	1	bernoulli(q	bernoulli(q	NOUN
asir-36063	206	2	)	)	PUNCT
asir-36063	206	3	,	,	PUNCT
asir-36063	206	4	independent	independent	ADJ
asir-36063	206	5	of	of	ADP
asir-36063	206	6	𝑊1	𝑊1	PROPN
asir-36063	206	7	,	,	PUNCT
asir-36063	206	8	…	…	PUNCT
asir-36063	206	9	,	,	PUNCT
asir-36063	206	10	𝑊𝑚.	𝑊𝑚.	NOUN
asir-36063	206	11	definition	definition	NOUN
asir-36063	206	12	a.4	a.4	X
asir-36063	206	13	.	.	PUNCT
asir-36063	207	1	we	we	PRON
asir-36063	207	2	define	define	VERB
asir-36063	207	3	:	:	PUNCT
asir-36063	207	4	𝑉(𝑚	𝑉(𝑚	ADJ
asir-36063	207	5	,	,	PUNCT
asir-36063	207	6	𝑝	𝑝	PROPN
asir-36063	207	7	,	,	PUNCT
asir-36063	207	8	𝑞	𝑞	PROPN
asir-36063	207	9	,	,	PUNCT
asir-36063	207	10	𝑡	𝑡	PROPN
asir-36063	207	11	,	,	PUNCT
asir-36063	207	12	𝑐	𝑐	NOUN
asir-36063	207	13	)	)	PUNCT
asir-36063	207	14	⁡=	⁡=	NOUN
asir-36063	207	15	(	(	PUNCT
asir-36063	207	16	𝑚	𝑚	X
asir-36063	207	17	(	(	PUNCT
asir-36063	207	18	𝑡	𝑡	X
asir-36063	207	19	+	+	NOUN
asir-36063	207	20	𝑐	𝑐	NOUN
asir-36063	207	21	)	)	PUNCT
asir-36063	207	22	𝑚	𝑚	NOUN
asir-36063	207	23	𝑛	𝑛	PROPN
asir-36063	207	24	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	207	25	)	)	PUNCT
asir-36063	207	26	)	)	PUNCT
asir-36063	208	1	(	(	PUNCT
asir-36063	208	2	𝑚	𝑚	X
asir-36063	208	3	𝑡	𝑡	PART
asir-36063	208	4	𝑚	𝑚	NOUN
asir-36063	208	5	𝑛	𝑛	DET
asir-36063	208	6	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	208	7	)	)	PUNCT
asir-36063	208	8	)	)	PUNCT
asir-36063	209	1	𝑝	𝑝	X
asir-36063	209	2	𝑡	𝑡	PROPN
asir-36063	209	3	𝑚	𝑚	PROPN
asir-36063	209	4	𝑛	𝑛	PRON
asir-36063	209	5	log⁡(𝑛)𝑞(𝑡+𝑐	log⁡(𝑛)𝑞(𝑡+𝑐	PROPN
asir-36063	209	6	)	)	PUNCT
asir-36063	209	7	𝑚	𝑚	ADP
asir-36063	209	8	𝑛	𝑛	DET
asir-36063	209	9	log⁡(𝑛)(1	log⁡(𝑛)(1	NOUN
asir-36063	209	10	−	−	PROPN
asir-36063	209	11	𝑝)𝑚−𝑡	𝑝)𝑚−𝑡	PROPN
asir-36063	209	12	𝑚	𝑚	PROPN
asir-36063	209	13	𝑛	𝑛	DET
asir-36063	209	14	log⁡(𝑛)(1	log⁡(𝑛)(1	ADJ
asir-36063	209	15	−	−	NOUN
asir-36063	209	16	𝑞)(𝑡+𝑐	𝑞)(𝑡+𝑐	NOUN
asir-36063	209	17	)	)	PUNCT
asir-36063	209	18	𝑚	𝑚	ADP
asir-36063	209	19	𝑛	𝑛	PROPN
asir-36063	209	20	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	209	21	)	)	PUNCT
asir-36063	209	22	(	(	PUNCT
asir-36063	209	23	a.10	a.10	PROPN
asir-36063	209	24	)	)	PUNCT
asir-36063	209	25	where	where	SCONJ
asir-36063	209	26	𝑐	𝑐	NOUN
asir-36063	209	27	=	=	SYM
asir-36063	209	28	𝑂(1	𝑂(1	ADP
asir-36063	209	29	)	)	PUNCT
asir-36063	209	30	.	.	PUNCT
asir-36063	210	1	we	we	PRON
asir-36063	210	2	also	also	ADV
asir-36063	210	3	define	define	VERB
asir-36063	210	4	the	the	DET
asir-36063	210	5	function	function	NOUN
asir-36063	210	6	:	:	PUNCT
asir-36063	211	1	𝑔(𝑎	𝑔(𝑎	ADJ
asir-36063	211	2	,	,	PUNCT
asir-36063	211	3	𝑏	𝑏	NOUN
asir-36063	211	4	,	,	PUNCT
asir-36063	211	5	𝑐	𝑐	NOUN
asir-36063	211	6	)	)	PUNCT
asir-36063	211	7	=	=	NOUN
asir-36063	211	8	(	(	PUNCT
asir-36063	211	9	𝑎	𝑎	X
asir-36063	211	10	+	+	NOUN
asir-36063	211	11	𝑏	𝑏	NOUN
asir-36063	211	12	)	)	PUNCT
asir-36063	211	13	−	−	NOUN
asir-36063	211	14	𝑐log⁡(𝑏	𝑐log⁡(𝑏	NOUN
asir-36063	211	15	)	)	PUNCT
asir-36063	212	1	−	−	PROPN
asir-36063	212	2	2√	2√	NUM
asir-36063	212	3	(	(	PUNCT
asir-36063	212	4	𝑐	𝑐	PROPN
asir-36063	212	5	2	2	NUM
asir-36063	212	6	)	)	PUNCT
asir-36063	212	7	2	2	NUM
asir-36063	212	8	+	+	CCONJ
asir-36063	212	9	𝑎𝑏	𝑎𝑏	PROPN
asir-36063	212	10	+	+	SYM
asir-36063	212	11	𝑐	𝑐	PROPN
asir-36063	212	12	2	2	NUM
asir-36063	212	13	log⁡(𝑎𝑏	log⁡(𝑎𝑏	NOUN
asir-36063	212	14	√	√	NUM
asir-36063	212	15	(	(	PUNCT
asir-36063	212	16	𝑐	𝑐	PROPN
asir-36063	212	17	2	2	NUM
asir-36063	212	18	)	)	PUNCT
asir-36063	212	19	2	2	NUM
asir-36063	213	1	+	+	NOUN
asir-36063	213	2	𝑎𝑏+	𝑎𝑏+	ADJ
asir-36063	213	3	𝑐	𝑐	NOUN
asir-36063	213	4	2	2	NUM
asir-36063	213	5	√	√	PROPN
asir-36063	213	6	(	(	PUNCT
asir-36063	213	7	𝑐	𝑐	PROPN
asir-36063	213	8	2	2	NUM
asir-36063	213	9	)	)	PUNCT
asir-36063	213	10	2	2	NUM
asir-36063	214	1	+	+	NOUN
asir-36063	214	2	𝑎𝑏−	𝑎𝑏−	X
asir-36063	214	3	𝑐	𝑐	NOUN
asir-36063	214	4	2	2	NUM
asir-36063	214	5	)	)	PUNCT
asir-36063	214	6	(	(	PUNCT
asir-36063	214	7	a.11	a.11	NOUN
asir-36063	214	8	)	)	PUNCT
asir-36063	214	9	then	then	ADV
asir-36063	214	10	we	we	PRON
asir-36063	214	11	have	have	VERB
asir-36063	214	12	the	the	DET
asir-36063	214	13	following	follow	VERB
asir-36063	214	14	results	result	NOUN
asir-36063	214	15	for	for	ADP
asir-36063	214	16	𝑇∗(𝑚	𝑇∗(𝑚	NOUN
asir-36063	214	17	,	,	PUNCT
asir-36063	214	18	𝑝	𝑝	NOUN
asir-36063	214	19	,	,	PUNCT
asir-36063	214	20	𝑞	𝑞	PROPN
asir-36063	214	21	,	,	PUNCT
asir-36063	214	22	𝑐	𝑐	NOUN
asir-36063	214	23	)	)	PUNCT
asir-36063	214	24	=	=	SYM
asir-36063	214	25	max𝑡>0	max𝑡>0	X
asir-36063	214	26	 	 	SPACE
asir-36063	214	27	𝑉(𝑚	𝑉(𝑚	PROPN
asir-36063	214	28	,	,	PUNCT
asir-36063	214	29	𝑝	𝑝	PROPN
asir-36063	214	30	,	,	PUNCT
asir-36063	214	31	𝑞	𝑞	PROPN
asir-36063	214	32	,	,	PUNCT
asir-36063	214	33	𝑡	𝑡	PROPN
asir-36063	214	34	,	,	PUNCT
asir-36063	214	35	𝑐	𝑐	NOUN
asir-36063	214	36	)	)	PUNCT
asir-36063	214	37	:	:	PUNCT
asir-36063	215	1	[	[	X
asir-36063	215	2	7	7	X
asir-36063	215	3	]	]	PUNCT
asir-36063	215	4	for	for	ADP
asir-36063	215	5	𝑚	𝑚	PROPN
asir-36063	215	6	∈	∈	PROPN
asir-36063	215	7	ℕ	ℕ	PROPN
asir-36063	215	8	and	and	CCONJ
asir-36063	215	9	∀𝑡	∀𝑡	NOUN
asir-36063	215	10	>	>	X
asir-36063	215	11	0	0	NUM
asir-36063	215	12	:	:	PUNCT
asir-36063	215	13	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	215	14	applied	apply	VERB
asir-36063	215	15	science	science	NOUN
asir-36063	215	16	and	and	CCONJ
asir-36063	215	17	innovative	innovative	ADJ
asir-36063	215	18	research	research	NOUN
asir-36063	215	19	vol	vol	NOUN
asir-36063	215	20	.	.	PROPN
asir-36063	215	21	8	8	NUM
asir-36063	215	22	,	,	PUNCT
asir-36063	215	23	no	no	INTJ
asir-36063	215	24	.	.	NOUN
asir-36063	215	25	1	1	NUM
asir-36063	215	26	,	,	PUNCT
asir-36063	215	27	2024	2024	NUM
asir-36063	215	28	73	73	NUM
asir-36063	215	29	published	publish	VERB
asir-36063	215	30	by	by	ADP
asir-36063	215	31	scholink	scholink	PROPN
asir-36063	215	32	inc	inc	PROPN
asir-36063	215	33	.	.	PROPN
asir-36063	215	34	−log⁡(𝑇∗(𝑚	−log⁡(𝑇∗(𝑚	PROPN
asir-36063	215	35	,	,	PUNCT
asir-36063	215	36	𝑝	𝑝	PROPN
asir-36063	215	37	,	,	PUNCT
asir-36063	215	38	𝑞	𝑞	PROPN
asir-36063	215	39	,	,	PUNCT
asir-36063	215	40	𝑐	𝑐	NOUN
asir-36063	215	41	)	)	PUNCT
asir-36063	215	42	)	)	PUNCT
asir-36063	215	43	≥	≥	X
asir-36063	215	44	𝑚	𝑚	X
asir-36063	215	45	𝑛	𝑛	PRON
asir-36063	215	46	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	215	47	)	)	PUNCT
asir-36063	215	48	∗	∗	PROPN
asir-36063	215	49	𝑔(𝑚	𝑔(𝑚	PROPN
asir-36063	215	50	,	,	PUNCT
asir-36063	215	51	𝑛	𝑛	PROPN
asir-36063	215	52	,	,	PUNCT
asir-36063	215	53	𝑐	𝑐	NOUN
asir-36063	215	54	)	)	PUNCT
asir-36063	215	55	−	−	PROPN
asir-36063	216	1	𝑜	𝑜	NOUN
asir-36063	216	2	(	(	PUNCT
asir-36063	216	3	𝑚	𝑚	PROPN
asir-36063	216	4	𝑛	𝑛	PROPN
asir-36063	216	5	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	216	6	)	)	PUNCT
asir-36063	216	7	)	)	PUNCT
asir-36063	217	1	∀𝑚	∀𝑚	PROPN
asir-36063	217	2	∈	∈	PROPN
asir-36063	217	3	ℕ	ℕ	PROPN
asir-36063	217	4	(	(	PUNCT
asir-36063	217	5	a.12	a.12	NOUN
asir-36063	217	6	)	)	PUNCT
asir-36063	217	7	in	in	ADP
asir-36063	217	8	the	the	DET
asir-36063	217	9	following	follow	VERB
asir-36063	217	10	proof	proof	NOUN
asir-36063	217	11	of	of	ADP
asir-36063	217	12	this	this	DET
asir-36063	217	13	lemma	lemma	PROPN
asir-36063	217	14	,	,	PUNCT
asir-36063	217	15	we	we	PRON
asir-36063	217	16	will	will	AUX
asir-36063	217	17	omit	omit	VERB
asir-36063	217	18	the	the	DET
asir-36063	217	19	ceiling	ceiling	NOUN
asir-36063	217	20	symbol	symbol	NOUN
asir-36063	217	21	for	for	ADP
asir-36063	217	22	clarity	clarity	NOUN
asir-36063	217	23	.	.	PUNCT
asir-36063	218	1	in	in	ADP
asir-36063	218	2	the	the	DET
asir-36063	218	3	case	case	NOUN
asir-36063	218	4	of	of	ADP
asir-36063	218	5	the	the	DET
asir-36063	218	6	directed	direct	VERB
asir-36063	218	7	stochastic	stochastic	ADJ
asir-36063	218	8	block	block	NOUN
asir-36063	218	9	model	model	NOUN
asir-36063	218	10	,	,	PUNCT
asir-36063	218	11	recall	recall	VERB
asir-36063	218	12	definition	definition	NOUN
asir-36063	218	13	a.3	a.3	PROPN
asir-36063	218	14	,	,	PUNCT
asir-36063	218	15	we	we	PRON
asir-36063	218	16	have	have	VERB
asir-36063	218	17	:	:	PUNCT
asir-36063	218	18	𝑇(𝑚	𝑇(𝑚	PROPN
asir-36063	218	19	,	,	PUNCT
asir-36063	218	20	𝑝	𝑝	PROPN
asir-36063	218	21	,	,	PUNCT
asir-36063	218	22	𝑞	𝑞	X
asir-36063	218	23	,	,	PUNCT
asir-36063	218	24	0	0	NUM
asir-36063	218	25	)	)	PUNCT
asir-36063	218	26	=	=	VERB
asir-36063	218	27	ℙ[𝑍	ℙ[𝑍	VERB
asir-36063	218	28	−𝑊	−𝑊	PROPN
asir-36063	218	29	≥	≥	NOUN
asir-36063	218	30	0	0	NUM
asir-36063	218	31	)	)	PUNCT
asir-36063	218	32	]	]	PUNCT
asir-36063	219	1	(	(	PUNCT
asir-36063	219	2	a.13	a.13	X
asir-36063	219	3	)	)	PUNCT
asir-36063	219	4	where	where	SCONJ
asir-36063	219	5	𝑍	𝑍	NOUN
asir-36063	219	6	is	be	AUX
asir-36063	219	7	a	a	DET
asir-36063	219	8	binomial⁡(𝑚	binomial⁡(𝑚	NOUN
asir-36063	219	9	,	,	PUNCT
asir-36063	219	10	𝑞	𝑞	NOUN
asir-36063	219	11	)	)	PUNCT
asir-36063	219	12	and	and	CCONJ
asir-36063	219	13	𝑊	𝑊	PROPN
asir-36063	219	14	is	be	AUX
asir-36063	219	15	a	a	DET
asir-36063	219	16	binomial	binomial	ADJ
asir-36063	219	17	(	(	PUNCT
asir-36063	219	18	𝑚	𝑚	NOUN
asir-36063	219	19	,	,	PUNCT
asir-36063	219	20	𝑝	𝑝	NOUN
asir-36063	219	21	)	)	PUNCT
asir-36063	219	22	,	,	PUNCT
asir-36063	219	23	𝑝	𝑝	NOUN
asir-36063	219	24	=	=	SYM
asir-36063	219	25	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	219	26	)	)	PUNCT
asir-36063	219	27	𝑛	𝑛	NOUN
asir-36063	219	28	,	,	PUNCT
asir-36063	219	29	𝑞	𝑞	X
asir-36063	219	30	=	=	PUNCT
asir-36063	219	31	𝑏log⁡(𝑛	𝑏log⁡(𝑛	PROPN
asir-36063	219	32	)	)	PUNCT
asir-36063	219	33	𝑛	𝑛	NOUN
asir-36063	219	34	.	.	PUNCT
asir-36063	220	1	we	we	PRON
asir-36063	220	2	can	can	AUX
asir-36063	220	3	re	re	VERB
asir-36063	220	4	-	-	VERB
asir-36063	220	5	write	write	ADJ
asir-36063	220	6	(	(	PUNCT
asir-36063	220	7	a.10	a.10	PROPN
asir-36063	220	8	)	)	PUNCT
asir-36063	220	9	into	into	ADP
asir-36063	220	10	:	:	PUNCT
asir-36063	220	11	𝑇(𝑚	𝑇(𝑚	PROPN
asir-36063	220	12	,	,	PUNCT
asir-36063	220	13	𝑝	𝑝	PROPN
asir-36063	220	14	,	,	PUNCT
asir-36063	220	15	𝑞	𝑞	X
asir-36063	220	16	,	,	PUNCT
asir-36063	220	17	0	0	NUM
asir-36063	220	18	)	)	PUNCT
asir-36063	220	19	=	=	PUNCT
asir-36063	221	1	∑	∑	PUNCT
asir-36063	221	2	 	 	SPACE
asir-36063	221	3	𝑚	𝑚	PROPN
asir-36063	221	4	𝑘1=0	𝑘1=0	PROPN
asir-36063	221	5	(	(	PUNCT
asir-36063	221	6	∑	∑	PART
asir-36063	221	7	 	 	SPACE
asir-36063	221	8	𝑚	𝑚	PROPN
asir-36063	221	9	𝑘2=𝑘1	𝑘2=𝑘1	PROPN
asir-36063	221	10	 	 	SPACE
asir-36063	221	11	ℙ(𝑍	ℙ(𝑍	NOUN
asir-36063	222	1	=	=	PUNCT
asir-36063	222	2	𝑘2))ℙ(𝑊	𝑘2))ℙ(𝑊	NUM
asir-36063	222	3	=	=	SYM
asir-36063	222	4	𝑘1	𝑘1	PROPN
asir-36063	222	5	)	)	PUNCT
asir-36063	222	6	(	(	PUNCT
asir-36063	222	7	a.14	a.14	X
asir-36063	222	8	)	)	PUNCT
asir-36063	222	9	where	where	SCONJ
asir-36063	222	10	each	each	DET
asir-36063	222	11	term	term	NOUN
asir-36063	222	12	in	in	ADP
asir-36063	222	13	the	the	DET
asir-36063	222	14	double	double	ADJ
asir-36063	222	15	summation	summation	NOUN
asir-36063	222	16	can	can	AUX
asir-36063	222	17	be	be	AUX
asir-36063	222	18	upper	upper	ADV
asir-36063	222	19	-	-	PUNCT
asir-36063	222	20	bounded	bound	VERB
asir-36063	222	21	by	by	ADP
asir-36063	222	22	𝑇∗(𝑚	𝑇∗(𝑚	PROPN
asir-36063	222	23	,	,	PUNCT
asir-36063	222	24	𝑝	𝑝	PROPN
asir-36063	222	25	,	,	PUNCT
asir-36063	222	26	𝑞	𝑞	X
asir-36063	222	27	,	,	PUNCT
asir-36063	222	28	0	0	NUM
asir-36063	222	29	)	)	PUNCT
asir-36063	222	30	.	.	PUNCT
asir-36063	223	1	using	use	VERB
asir-36063	223	2	𝑐	𝑐	PROPN
asir-36063	223	3	=	=	SYM
asir-36063	223	4	0	0	NUM
asir-36063	223	5	,	,	PUNCT
asir-36063	223	6	we	we	PRON
asir-36063	223	7	have	have	VERB
asir-36063	223	8	𝑇(𝑚	𝑇(𝑚	PROPN
asir-36063	223	9	,	,	PUNCT
asir-36063	223	10	𝑝	𝑝	PROPN
asir-36063	223	11	,	,	PUNCT
asir-36063	223	12	𝑞	𝑞	PROPN
asir-36063	223	13	,	,	PUNCT
asir-36063	223	14	0)⁡≤	0)⁡≤	NUM
asir-36063	223	15	𝑚2𝑇∗(𝑚	𝑚2𝑇∗(𝑚	PROPN
asir-36063	223	16	,	,	PUNCT
asir-36063	223	17	𝑝	𝑝	PROPN
asir-36063	223	18	,	,	PUNCT
asir-36063	223	19	𝑞	𝑞	X
asir-36063	223	20	,	,	PUNCT
asir-36063	223	21	0	0	NUM
asir-36063	223	22	)	)	PUNCT
asir-36063	223	23	−log⁡(𝑇(𝑚	−log⁡(𝑇(𝑚	PROPN
asir-36063	223	24	,	,	PUNCT
asir-36063	223	25	𝑝	𝑝	PROPN
asir-36063	223	26	,	,	PUNCT
asir-36063	223	27	𝑞	𝑞	PROPN
asir-36063	223	28	,	,	PUNCT
asir-36063	223	29	0))⁡≥	0))⁡≥	PROPN
asir-36063	223	30	−2log⁡(𝑚	−2log⁡(𝑚	PROPN
asir-36063	223	31	)	)	PUNCT
asir-36063	223	32	−	−	PROPN
asir-36063	224	1	log⁡(𝑇∗(𝑚	log⁡(𝑇∗(𝑚	PROPN
asir-36063	224	2	,	,	PUNCT
asir-36063	224	3	𝑝	𝑝	PROPN
asir-36063	224	4	,	,	PUNCT
asir-36063	224	5	𝑞	𝑞	X
asir-36063	224	6	,	,	PUNCT
asir-36063	224	7	0	0	NUM
asir-36063	224	8	)	)	PUNCT
asir-36063	224	9	)	)	PUNCT
asir-36063	224	10	⁡≥	⁡≥	PUNCT
asir-36063	224	11	−2log⁡(𝑚	−2log⁡(𝑚	X
asir-36063	224	12	)	)	PUNCT
asir-36063	224	13	+	+	CCONJ
asir-36063	224	14	2𝑚	2𝑚	NUM
asir-36063	224	15	𝑛	𝑛	ADP
asir-36063	224	16	(	(	PUNCT
asir-36063	224	17	𝑎+𝑏	𝑎+𝑏	NUM
asir-36063	224	18	2	2	NUM
asir-36063	224	19	−	−	NUM
asir-36063	224	20	√𝑎𝑏	√𝑎𝑏	NOUN
asir-36063	224	21	)	)	PUNCT
asir-36063	224	22	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	224	23	)	)	PUNCT
asir-36063	224	24	(	(	PUNCT
asir-36063	224	25	a.15	a.15	NOUN
asir-36063	224	26	)	)	PUNCT
asir-36063	224	27	as	as	ADV
asir-36063	224	28	long	long	ADV
asir-36063	224	29	as	as	ADP
asir-36063	224	30	𝑚	𝑚	X
asir-36063	224	31	𝑛	𝑛	PROPN
asir-36063	224	32	>	>	X
asir-36063	224	33	log⁡log⁡(𝑛	log⁡log⁡(𝑛	PROPN
asir-36063	224	34	)	)	PUNCT
asir-36063	224	35	and	and	CCONJ
asir-36063	224	36	𝑚	𝑚	X
asir-36063	224	37	≤	≤	NUM
asir-36063	224	38	𝑛2	𝑛2	NOUN
asir-36063	224	39	4	4	NUM
asir-36063	224	40	,	,	PUNCT
asir-36063	224	41	then	then	ADV
asir-36063	224	42	we	we	PRON
asir-36063	224	43	have	have	VERB
asir-36063	224	44	log⁡(𝑚	log⁡(𝑚	NUM
asir-36063	224	45	)	)	PUNCT
asir-36063	225	1	=	=	SYM
asir-36063	225	2	𝑜	𝑜	X
asir-36063	225	3	(	(	PUNCT
asir-36063	225	4	𝑚	𝑚	PROPN
asir-36063	225	5	𝑛	𝑛	PROPN
asir-36063	225	6	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	225	7	)	)	PUNCT
asir-36063	225	8	)	)	PUNCT
asir-36063	225	9	.	.	PUNCT
asir-36063	226	1	hence	hence	ADV
asir-36063	226	2	,	,	PUNCT
asir-36063	226	3	−log⁡(𝑇(𝑚	−log⁡(𝑇(𝑚	PROPN
asir-36063	226	4	,	,	PUNCT
asir-36063	226	5	𝑝	𝑝	PROPN
asir-36063	226	6	,	,	PUNCT
asir-36063	226	7	𝑞	𝑞	X
asir-36063	226	8	,	,	PUNCT
asir-36063	226	9	0	0	NUM
asir-36063	226	10	)	)	PUNCT
asir-36063	226	11	)	)	PUNCT
asir-36063	226	12	≥	≥	NOUN
asir-36063	226	13	2𝑚	2𝑚	NUM
asir-36063	226	14	𝑛	𝑛	PROPN
asir-36063	226	15	(	(	PUNCT
asir-36063	226	16	𝑎+𝑏	𝑎+𝑏	NUM
asir-36063	226	17	2	2	NUM
asir-36063	226	18	−	−	NUM
asir-36063	226	19	√𝑎𝑏	√𝑎𝑏	NOUN
asir-36063	226	20	)	)	PUNCT
asir-36063	226	21	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	226	22	)	)	PUNCT
asir-36063	227	1	−	−	PROPN
asir-36063	227	2	𝑜	𝑜	NOUN
asir-36063	227	3	(	(	PUNCT
asir-36063	227	4	𝑚	𝑚	PROPN
asir-36063	227	5	𝑛	𝑛	PROPN
asir-36063	227	6	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	227	7	)	)	PUNCT
asir-36063	227	8	)	)	PUNCT
asir-36063	227	9	(	(	PUNCT
asir-36063	227	10	a.16	a.16	PROPN
asir-36063	227	11	)	)	PUNCT
asir-36063	227	12	in	in	ADP
asir-36063	227	13	this	this	DET
asir-36063	227	14	case	case	NOUN
asir-36063	227	15	,	,	PUNCT
asir-36063	227	16	𝑇(𝑚	𝑇(𝑚	PROPN
asir-36063	227	17	,	,	PUNCT
asir-36063	227	18	𝑝	𝑝	PROPN
asir-36063	227	19	,	,	PUNCT
asir-36063	227	20	𝑞	𝑞	X
asir-36063	227	21	,	,	PUNCT
asir-36063	227	22	0	0	NUM
asir-36063	227	23	)	)	PUNCT
asir-36063	227	24	is	be	AUX
asir-36063	227	25	equivalent	equivalent	ADJ
asir-36063	227	26	to	to	ADP
asir-36063	227	27	ℙ(bin⁡	ℙ(bin⁡	PROPN
asir-36063	227	28	(	(	PUNCT
asir-36063	227	29	𝑛	𝑛	PROPN
asir-36063	227	30	2	2	NUM
asir-36063	227	31	,	,	PUNCT
asir-36063	227	32	𝑞	𝑞	PROPN
asir-36063	227	33	)	)	PUNCT
asir-36063	227	34	>	>	X
asir-36063	227	35	bin⁡	bin⁡	PROPN
asir-36063	227	36	(	(	PUNCT
asir-36063	227	37	𝑛	𝑛	PROPN
asir-36063	227	38	2	2	NUM
asir-36063	227	39	,	,	PUNCT
asir-36063	227	40	𝑝	𝑝	NOUN
asir-36063	227	41	)	)	PUNCT
asir-36063	227	42	)	)	PUNCT
asir-36063	227	43	,	,	PUNCT
asir-36063	227	44	and	and	CCONJ
asir-36063	227	45	continuing	continue	VERB
asir-36063	227	46	(	(	PUNCT
asir-36063	227	47	a.8	a.8	NOUN
asir-36063	227	48	)	)	PUNCT
asir-36063	227	49	,	,	PUNCT
asir-36063	227	50	as	as	SCONJ
asir-36063	227	51	𝑛	𝑛	DET
asir-36063	227	52	approaches	approach	NOUN
asir-36063	227	53	infinity	infinity	NOUN
asir-36063	227	54	,	,	PUNCT
asir-36063	227	55	we	we	PRON
asir-36063	227	56	get	get	VERB
asir-36063	227	57	:	:	PUNCT
asir-36063	227	58	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	227	59	∈	∈	PROPN
asir-36063	227	60	𝐵1	𝐵1	NOUN
asir-36063	227	61	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	227	62	)	)	PUNCT
asir-36063	227	63	)	)	PUNCT
asir-36063	228	1	=	=	PUNCT
asir-36063	228	2	𝑛ℙ(𝑑−	𝑛ℙ(𝑑−	NOUN
asir-36063	228	3	𝑅	𝑅	PROPN
asir-36063	228	4	>	>	X
asir-36063	228	5	𝑑+	𝑑+	PROPN
asir-36063	228	6	𝑅	𝑅	PROPN
asir-36063	228	7	)	)	PUNCT
asir-36063	228	8	=	=	SYM
asir-36063	228	9	𝑛ℙ(bin⁡	𝑛ℙ(bin⁡	PROPN
asir-36063	228	10	(	(	PUNCT
asir-36063	228	11	𝑛	𝑛	PROPN
asir-36063	228	12	2	2	NUM
asir-36063	228	13	,	,	PUNCT
asir-36063	228	14	𝑞	𝑞	PROPN
asir-36063	228	15	)	)	PUNCT
asir-36063	228	16	>	>	X
asir-36063	228	17	bin⁡	bin⁡	PROPN
asir-36063	228	18	(	(	PUNCT
asir-36063	228	19	𝑛	𝑛	PROPN
asir-36063	228	20	2	2	NUM
asir-36063	228	21	,	,	PUNCT
asir-36063	228	22	𝑝	𝑝	NOUN
asir-36063	228	23	)	)	PUNCT
asir-36063	228	24	)	)	PUNCT
asir-36063	229	1	=	=	SYM
asir-36063	229	2	𝑛	𝑛	DET
asir-36063	229	3	1−	1−	NUM
asir-36063	229	4	(	(	PUNCT
asir-36063	229	5	√𝑎−√𝑏	√𝑎−√𝑏	X
asir-36063	229	6	√2	√2	PROPN
asir-36063	229	7	)	)	PUNCT
asir-36063	229	8	2	2	NUM
asir-36063	229	9	+	+	NOUN
asir-36063	229	10	𝑜(1	𝑜(1	NUM
asir-36063	229	11	)	)	PUNCT
asir-36063	229	12	(	(	PUNCT
asir-36063	229	13	a.17	a.17	X
asir-36063	229	14	)	)	PUNCT
asir-36063	229	15	therefore	therefore	ADV
asir-36063	229	16	,	,	PUNCT
asir-36063	229	17	when	when	SCONJ
asir-36063	229	18	√𝑎	√𝑎	ADJ
asir-36063	229	19	−	−	NOUN
asir-36063	229	20	√𝑏	√𝑏	VERB
asir-36063	229	21	<	<	X
asir-36063	229	22	√2	√2	PROPN
asir-36063	229	23	,	,	PUNCT
asir-36063	229	24	ℙ(∃𝑢	ℙ(∃𝑢	PROPN
asir-36063	229	25	∈	∈	PROPN
asir-36063	229	26	𝐵1	𝐵1	NOUN
asir-36063	229	27	𝑅(𝐺	𝑅(𝐺	PROPN
asir-36063	229	28	)	)	PUNCT
asir-36063	229	29	)	)	PUNCT
asir-36063	230	1	=	=	SYM
asir-36063	230	2	1	1	NUM
asir-36063	230	3	−	−	NOUN
asir-36063	230	4	𝑜(1	𝑜(1	NOUN
asir-36063	230	5	)	)	PUNCT
asir-36063	230	6	,	,	PUNCT
asir-36063	230	7	there	there	PRON
asir-36063	230	8	exists	exist	VERB
asir-36063	230	9	a	a	DET
asir-36063	230	10	bad	bad	ADJ
asir-36063	230	11	vertex	vertex	NOUN
asir-36063	230	12	and	and	CCONJ
asir-36063	230	13	exact	exact	ADJ
asir-36063	230	14	recovery	recovery	NOUN
asir-36063	230	15	is	be	AUX
asir-36063	230	16	unachievable	unachievable	ADJ
asir-36063	230	17	.	.	PUNCT
asir-36063	231	1	b	b	X
asir-36063	231	2	proof	proof	NOUN
asir-36063	231	3	for	for	ADP
asir-36063	231	4	theorem	theorem	VERB
asir-36063	231	5	3.3	3.3	NUM
asir-36063	231	6	proof	proof	NOUN
asir-36063	231	7	for	for	ADP
asir-36063	231	8	lemma	lemma	PROPN
asir-36063	231	9	3.4	3.4	NUM
asir-36063	231	10	let	let	VERB
asir-36063	231	11	𝒈	𝒈	PROPN
asir-36063	231	12	=	=	PUNCT
asir-36063	231	13	(	(	PUNCT
asir-36063	231	14	1	1	NUM
asir-36063	231	15	,	,	PUNCT
asir-36063	231	16	…	…	PUNCT
asir-36063	231	17	,	,	PUNCT
asir-36063	231	18	1	1	NUM
asir-36063	231	19	,	,	PUNCT
asir-36063	231	20	−1	−1	NOUN
asir-36063	231	21	,	,	PUNCT
asir-36063	231	22	…	…	PUNCT
asir-36063	231	23	,	,	PUNCT
asir-36063	231	24	−1,1	−1,1	NOUN
asir-36063	231	25	,	,	PUNCT
asir-36063	231	26	.	.	PUNCT
asir-36063	231	27	.	.	PUNCT
asir-36063	232	1	,	,	PUNCT
asir-36063	232	2	1	1	NUM
asir-36063	232	3	,	,	PUNCT
asir-36063	232	4	−1	−1	NOUN
asir-36063	232	5	,	,	PUNCT
asir-36063	232	6	…	…	PUNCT
asir-36063	232	7	,	,	PUNCT
asir-36063	232	8	−1	−1	NOUN
asir-36063	232	9	)	)	PUNCT
asir-36063	232	10	.	.	PUNCT
asir-36063	233	1	without	without	ADP
asir-36063	233	2	loss	loss	NOUN
asir-36063	233	3	of	of	ADP
asir-36063	233	4	generality	generality	NOUN
asir-36063	233	5	.	.	PUNCT
asir-36063	234	1	by	by	ADP
asir-36063	234	2	kkt	kkt	PROPN
asir-36063	234	3	condition	condition	PROPN
asir-36063	234	4	,	,	PUNCT
asir-36063	234	5	we	we	PRON
asir-36063	234	6	obtain	obtain	VERB
asir-36063	234	7	a	a	DET
asir-36063	234	8	sufficient	sufficient	ADJ
asir-36063	234	9	condition	condition	NOUN
asir-36063	234	10	for	for	ADP
asir-36063	234	11	𝒈𝒈⊤	𝒈𝒈⊤	NOUN
asir-36063	234	12	to	to	PART
asir-36063	234	13	be	be	AUX
asir-36063	234	14	the	the	DET
asir-36063	234	15	solution	solution	NOUN
asir-36063	234	16	of	of	ADP
asir-36063	234	17	the	the	DET
asir-36063	234	18	sdp	sdp	NOUN
asir-36063	234	19	relaxation	relaxation	NOUN
asir-36063	234	20	(	(	PUNCT
asir-36063	234	21	2.5	2.5	NUM
asir-36063	234	22	)	)	PUNCT
asir-36063	234	23	.	.	PUNCT
asir-36063	235	1	therefore	therefore	ADV
asir-36063	235	2	,	,	PUNCT
asir-36063	235	3	we	we	PRON
asir-36063	235	4	have	have	VERB
asir-36063	235	5	𝚲	𝚲	NOUN
asir-36063	235	6	≽	≽	NOUN
asir-36063	235	7	0	0	NUM
asir-36063	235	8	,	,	PUNCT
asir-36063	235	9	and	and	CCONJ
asir-36063	235	10	𝒈𝒈⊤	𝒈𝒈⊤	NOUN
asir-36063	235	11	is	be	AUX
asir-36063	235	12	guaranteed	guarantee	VERB
asir-36063	235	13	to	to	PART
asir-36063	235	14	be	be	AUX
asir-36063	235	15	the	the	DET
asir-36063	235	16	optimal	optimal	ADJ
asir-36063	235	17	solution	solution	NOUN
asir-36063	235	18	to	to	ADP
asir-36063	235	19	sdp	sdp	NOUN
asir-36063	235	20	relaxation	relaxation	NOUN
asir-36063	235	21	(	(	PUNCT
asir-36063	235	22	2.5	2.5	NUM
asir-36063	235	23	)	)	PUNCT
asir-36063	235	24	if	if	SCONJ
asir-36063	235	25	:	:	PUNCT
asir-36063	235	26	1	1	X
asir-36063	235	27	.	.	X
asir-36063	235	28	𝒈𝒈⊤	𝒈𝒈⊤	NOUN
asir-36063	235	29	is	be	AUX
asir-36063	235	30	a	a	DET
asir-36063	235	31	solution	solution	NOUN
asir-36063	235	32	to	to	ADP
asir-36063	235	33	the	the	DET
asir-36063	235	34	primal	primal	ADJ
asir-36063	235	35	problem	problem	NOUN
asir-36063	235	36	,	,	PUNCT
asir-36063	235	37	2	2	X
asir-36063	235	38	.	.	X
asir-36063	235	39	there	there	PRON
asir-36063	235	40	exists	exist	VERB
asir-36063	235	41	a	a	DET
asir-36063	235	42	matrix	matrix	NOUN
asir-36063	235	43	𝒀	𝒀	NOUN
asir-36063	235	44	feasible	feasible	ADJ
asir-36063	235	45	for	for	ADP
asir-36063	235	46	the	the	DET
asir-36063	235	47	dual	dual	ADJ
asir-36063	235	48	problem	problem	NOUN
asir-36063	235	49	such	such	ADJ
asir-36063	235	50	that	that	PRON
asir-36063	235	51	tr⁡((2𝑨∗	tr⁡((2𝑨∗	PROPN
asir-36063	235	52	−	−	PROPN
asir-36063	235	53	𝑱2𝑛)𝒈𝒈	𝑱2𝑛)𝒈𝒈	PROPN
asir-36063	235	54	⊤	⊤	PROPN
asir-36063	235	55	)	)	PUNCT
asir-36063	235	56	=	=	SYM
asir-36063	235	57	tr⁡(𝒀	tr⁡(𝒀	NOUN
asir-36063	235	58	)	)	PUNCT
asir-36063	235	59	.	.	PUNCT
asir-36063	236	1	the	the	DET
asir-36063	236	2	first	first	ADJ
asir-36063	236	3	condition	condition	NOUN
asir-36063	236	4	is	be	AUX
asir-36063	236	5	already	already	ADV
asir-36063	236	6	satisfied	satisfied	ADJ
asir-36063	236	7	by	by	ADP
asir-36063	236	8	the	the	DET
asir-36063	236	9	given	give	VERB
asir-36063	236	10	background	background	NOUN
asir-36063	236	11	,	,	PUNCT
asir-36063	236	12	then	then	ADV
asir-36063	236	13	we	we	PRON
asir-36063	236	14	need	need	VERB
asir-36063	236	15	to	to	PART
asir-36063	236	16	find	find	VERB
asir-36063	236	17	a	a	DET
asir-36063	236	18	𝒀	𝒀	NOUN
asir-36063	236	19	(	(	PUNCT
asir-36063	236	20	also	also	ADV
asir-36063	236	21	known	know	VERB
asir-36063	236	22	as	as	ADP
asir-36063	236	23	dual	dual	ADJ
asir-36063	236	24	certificate	certificate	NOUN
asir-36063	236	25	)	)	PUNCT
asir-36063	236	26	that	that	PRON
asir-36063	236	27	would	would	AUX
asir-36063	236	28	satisfy	satisfy	VERB
asir-36063	236	29	the	the	DET
asir-36063	236	30	second	second	ADJ
asir-36063	236	31	condition	condition	NOUN
asir-36063	236	32	.	.	PUNCT
asir-36063	237	1	we	we	PRON
asir-36063	237	2	can	can	AUX
asir-36063	237	3	use	use	VERB
asir-36063	237	4	𝑪	𝑪	PROPN
asir-36063	237	5	to	to	PART
asir-36063	237	6	substitute	substitute	VERB
asir-36063	237	7	2𝑨∗	2𝑨∗	NOUN
asir-36063	237	8	−	−	PROPN
asir-36063	237	9	𝑱2𝑛	𝑱2𝑛	PROPN
asir-36063	237	10	,	,	PUNCT
asir-36063	237	11	then	then	ADV
asir-36063	237	12	we	we	PRON
asir-36063	237	13	have	have	VERB
asir-36063	237	14	:	:	PUNCT
asir-36063	237	15	(	(	PUNCT
asir-36063	237	16	𝑪𝒈𝒈⁡⊤)𝑖𝑖	𝑪𝒈𝒈⁡⊤)𝑖𝑖	PROPN
asir-36063	237	17	=	=	SYM
asir-36063	237	18	⁡correct⁡edges⁡	⁡correct⁡edges⁡	X
asir-36063	237	19	+	+	CCONJ
asir-36063	237	20	⁡correct⁡non	⁡correct⁡non	X
asir-36063	237	21	−	−	NOUN
asir-36063	237	22	edges⁡incorrect	edges⁡incorrect	ADJ
asir-36063	237	23	edges	edge	NOUN
asir-36063	237	24	incorrect	incorrect	ADJ
asir-36063	237	25	non	non	ADJ
asir-36063	237	26	-	-	ADJ
asir-36063	237	27	edges	edges	ADJ
asir-36063	237	28	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡=	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡=	NOUN
asir-36063	237	29	(	(	PUNCT
asir-36063	237	30	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	237	31	+	+	NOUN
asir-36063	237	32	)	)	PUNCT
asir-36063	237	33	𝑖𝑖	𝑖𝑖	X
asir-36063	238	1	+	+	CCONJ
asir-36063	238	2	(	(	PUNCT
asir-36063	238	3	𝑛	𝑛	PROPN
asir-36063	238	4	2	2	NUM
asir-36063	238	5	−	−	NOUN
asir-36063	238	6	(	(	PUNCT
asir-36063	238	7	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	238	8	−)𝑖𝑖	−)𝑖𝑖	NOUN
asir-36063	238	9	)	)	PUNCT
asir-36063	238	10	−	−	PROPN
asir-36063	239	1	(	(	PUNCT
asir-36063	239	2	𝑛	𝑛	PROPN
asir-36063	239	3	2	2	NUM
asir-36063	239	4	−	−	NOUN
asir-36063	239	5	1	1	NUM
asir-36063	239	6	−	−	PROPN
asir-36063	239	7	(	(	PUNCT
asir-36063	239	8	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	239	9	+	+	NOUN
asir-36063	239	10	)	)	PUNCT
asir-36063	239	11	𝑖𝑖	𝑖𝑖	ADP
asir-36063	239	12	)	)	PUNCT
asir-36063	239	13	−	−	PROPN
asir-36063	240	1	(	(	PUNCT
asir-36063	240	2	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	240	3	−)𝑖𝑖	−)𝑖𝑖	PUNCT
asir-36063	240	4	+	+	CCONJ
asir-36063	240	5	1	1	NUM
asir-36063	240	6	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡=	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡=	NOUN
asir-36063	240	7	2((𝑫𝐶	2((𝑫𝐶	PROPN
asir-36063	241	1	+	+	NOUN
asir-36063	241	2	)	)	PUNCT
asir-36063	242	1	𝑖𝑖	𝑖𝑖	NOUN
asir-36063	242	2	−	−	PROPN
asir-36063	243	1	(	(	PUNCT
asir-36063	243	2	𝑫𝐶	𝑫𝐶	PROPN
asir-36063	243	3	−)𝑖𝑖	−)𝑖𝑖	PROPN
asir-36063	243	4	)	)	PUNCT
asir-36063	244	1	+	+	CCONJ
asir-36063	244	2	1	1	NUM
asir-36063	244	3	(	(	PUNCT
asir-36063	244	4	b.1	b.1	NOUN
asir-36063	244	5	)	)	PUNCT
asir-36063	244	6	for	for	ADP
asir-36063	244	7	𝑖	𝑖	PRON
asir-36063	244	8	∈	∈	PROPN
asir-36063	245	1	[	[	X
asir-36063	245	2	𝑛	𝑛	NOUN
asir-36063	245	3	+	+	X
asir-36063	245	4	1,2𝑛	1,2𝑛	NUM
asir-36063	245	5	]	]	PUNCT
asir-36063	245	6	,	,	PUNCT
asir-36063	245	7	we	we	PRON
asir-36063	245	8	let	let	VERB
asir-36063	245	9	𝑗	𝑗	NOUN
asir-36063	245	10	=	=	PUNCT
asir-36063	245	11	𝑖	𝑖	SYM
asir-36063	245	12	−	−	PROPN
asir-36063	245	13	𝑛	𝑛	PROPN
asir-36063	245	14	,	,	PUNCT
asir-36063	245	15	then	then	ADV
asir-36063	245	16	we	we	PRON
asir-36063	245	17	have	have	VERB
asir-36063	245	18	:	:	PUNCT
asir-36063	245	19	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	245	20	applied	apply	VERB
asir-36063	245	21	science	science	NOUN
asir-36063	245	22	and	and	CCONJ
asir-36063	245	23	innovative	innovative	ADJ
asir-36063	245	24	research	research	NOUN
asir-36063	245	25	vol	vol	NOUN
asir-36063	245	26	.	.	PROPN
asir-36063	246	1	8	8	NUM
asir-36063	246	2	,	,	PUNCT
asir-36063	246	3	no	no	INTJ
asir-36063	246	4	.	.	NOUN
asir-36063	246	5	1	1	NUM
asir-36063	246	6	,	,	PUNCT
asir-36063	246	7	2024	2024	NUM
asir-36063	246	8	74	74	NUM
asir-36063	246	9	published	publish	VERB
asir-36063	246	10	by	by	ADP
asir-36063	246	11	scholink	scholink	PROPN
asir-36063	246	12	inc	inc	PROPN
asir-36063	246	13	.	.	PROPN
asir-36063	247	1	(	(	PUNCT
asir-36063	247	2	𝑪𝒈𝒈⊤)𝑖𝑖	𝑪𝒈𝒈⊤)𝑖𝑖	NOUN
asir-36063	247	3	=	=	PUNCT
asir-36063	247	4	2((𝑫𝑅	2((𝑫𝑅	NUM
asir-36063	248	1	+	+	ADJ
asir-36063	248	2	)	)	PUNCT
asir-36063	248	3	𝑗𝑗	𝑗𝑗	ADP
asir-36063	248	4	−	−	PROPN
asir-36063	248	5	(	(	PUNCT
asir-36063	248	6	𝑫𝑅	𝑫𝑅	PROPN
asir-36063	248	7	−)𝑗𝑗	−)𝑗𝑗	NUM
asir-36063	248	8	)	)	PUNCT
asir-36063	248	9	+	+	CCONJ
asir-36063	248	10	1	1	NUM
asir-36063	248	11	(	(	PUNCT
asir-36063	248	12	b.2	b.2	VERB
asir-36063	248	13	)	)	PUNCT
asir-36063	248	14	therefore	therefore	ADV
asir-36063	248	15	,	,	PUNCT
asir-36063	248	16	we	we	PRON
asir-36063	248	17	get	get	VERB
asir-36063	248	18	tr⁡(𝑪𝒈𝒈⊤	tr⁡(𝑪𝒈𝒈⊤	ADV
asir-36063	248	19	)	)	PUNCT
asir-36063	248	20	=	=	NOUN
asir-36063	248	21	tr⁡(2(𝑫𝐶	tr⁡(2(𝑫𝐶	ADJ
asir-36063	248	22	+	+	CCONJ
asir-36063	248	23	−𝑫𝐶	−𝑫𝐶	PROPN
asir-36063	248	24	−	−	NOUN
asir-36063	248	25	)	)	PUNCT
asir-36063	248	26	+	+	PUNCT
asir-36063	249	1	𝑰𝑛	𝑰𝑛	ADJ
asir-36063	249	2	)	)	PUNCT
asir-36063	249	3	+	+	CCONJ
asir-36063	249	4	tr⁡(2(𝑫𝑅	tr⁡(2(𝑫𝑅	NUM
asir-36063	249	5	+	+	CCONJ
asir-36063	249	6	−	−	PROPN
asir-36063	249	7	𝑫𝑅	𝑫𝑅	PROPN
asir-36063	249	8	−	−	NUM
asir-36063	249	9	)	)	PUNCT
asir-36063	249	10	+	+	PUNCT
asir-36063	250	1	𝑰𝑛	𝑰𝑛	ADJ
asir-36063	250	2	)	)	PUNCT
asir-36063	250	3	,	,	PUNCT
asir-36063	250	4	and	and	CCONJ
asir-36063	250	5	we	we	PRON
asir-36063	250	6	are	be	AUX
asir-36063	250	7	able	able	ADJ
asir-36063	250	8	to	to	PART
asir-36063	250	9	find	find	VERB
asir-36063	250	10	a	a	DET
asir-36063	250	11	matrix	matrix	NOUN
asir-36063	250	12	𝒀	𝒀	NOUN
asir-36063	250	13	that	that	PRON
asir-36063	250	14	is	be	AUX
asir-36063	250	15	feasible	feasible	ADJ
asir-36063	250	16	for	for	ADP
asir-36063	250	17	the	the	DET
asir-36063	250	18	dual	dual	ADJ
asir-36063	250	19	problem	problem	NOUN
asir-36063	250	20	and	and	CCONJ
asir-36063	250	21	satisfies	satisfy	VERB
asir-36063	250	22	the	the	DET
asir-36063	250	23	proposed	propose	VERB
asir-36063	250	24	condition	condition	NOUN
asir-36063	250	25	:	:	PUNCT
asir-36063	251	1	𝒀	𝒀	NOUN
asir-36063	251	2	=	=	PUNCT
asir-36063	251	3	[	[	PUNCT
asir-36063	251	4	2(𝑫𝐶	2(𝑫𝐶	NUM
asir-36063	251	5	+	+	NUM
asir-36063	251	6	−𝑫𝐶	−𝑫𝐶	PROPN
asir-36063	251	7	−	−	NOUN
asir-36063	251	8	)	)	PUNCT
asir-36063	252	1	+	+	CCONJ
asir-36063	252	2	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	252	3	0	0	NUM
asir-36063	252	4	0	0	NUM
asir-36063	252	5	2(𝑫𝑅	2(𝑫𝑅	NUM
asir-36063	252	6	+	+	CCONJ
asir-36063	252	7	−𝑫𝑅	−𝑫𝑅	ADJ
asir-36063	252	8	−	−	NOUN
asir-36063	252	9	)	)	PUNCT
asir-36063	253	1	+	+	CCONJ
asir-36063	253	2	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	253	3	]	]	PUNCT
asir-36063	253	4	.	.	PUNCT
asir-36063	254	1	as	as	ADP
asir-36063	254	2	a	a	DET
asir-36063	254	3	result	result	NOUN
asir-36063	254	4	,	,	PUNCT
asir-36063	254	5	if	if	SCONJ
asir-36063	254	6	𝚲	𝚲	PROPN
asir-36063	254	7	≽	≽	NOUN
asir-36063	254	8	0	0	NUM
asir-36063	254	9	,	,	PUNCT
asir-36063	254	10	then	then	ADV
asir-36063	254	11	𝒈𝒈⊤	𝒈𝒈⊤	ADV
asir-36063	254	12	is	be	AUX
asir-36063	254	13	the	the	DET
asir-36063	254	14	optimal	optimal	ADJ
asir-36063	254	15	solution	solution	NOUN
asir-36063	254	16	to	to	ADP
asir-36063	254	17	the	the	DET
asir-36063	254	18	sdp	sdp	NOUN
asir-36063	254	19	.	.	PUNCT
asir-36063	255	1	in	in	ADP
asir-36063	255	2	addition	addition	NOUN
asir-36063	255	3	,	,	PUNCT
asir-36063	255	4	𝜆2(𝚲	𝜆2(𝚲	PROPN
asir-36063	255	5	)	)	PUNCT
asir-36063	255	6	>	>	SYM
asir-36063	255	7	0	0	NUM
asir-36063	255	8	ensures	ensure	VERB
asir-36063	255	9	that	that	SCONJ
asir-36063	255	10	𝒈𝒈⊤	𝒈𝒈⊤	NOUN
asir-36063	255	11	is	be	AUX
asir-36063	255	12	the	the	DET
asir-36063	255	13	only	only	ADJ
asir-36063	255	14	solution	solution	NOUN
asir-36063	255	15	to	to	ADP
asir-36063	255	16	sdp	sdp	NOUN
asir-36063	255	17	.	.	PUNCT
asir-36063	256	1	imagine	imagine	VERB
asir-36063	256	2	there	there	PRON
asir-36063	256	3	is	be	VERB
asir-36063	256	4	another	another	DET
asir-36063	256	5	optimal	optimal	ADJ
asir-36063	256	6	solution	solution	NOUN
asir-36063	256	7	𝑿∗	𝑿∗	NUM
asir-36063	256	8	to	to	ADP
asir-36063	256	9	the	the	DET
asir-36063	256	10	sdp	sdp	NOUN
asir-36063	256	11	,	,	PUNCT
asir-36063	256	12	then	then	ADV
asir-36063	256	13	we	we	PRON
asir-36063	256	14	get	get	VERB
asir-36063	256	15	tr⁡(𝑿′𝚲	tr⁡(𝑿′𝚲	ADP
asir-36063	256	16	)	)	PUNCT
asir-36063	257	1	=	=	SYM
asir-36063	257	2	0	0	NUM
asir-36063	257	3	by	by	ADP
asir-36063	257	4	complementary	complementary	ADJ
asir-36063	257	5	slackness	slackness	NOUN
asir-36063	257	6	.	.	PUNCT
asir-36063	258	1	by	by	ADP
asir-36063	258	2	assumption	assumption	NOUN
asir-36063	258	3	,	,	PUNCT
asir-36063	258	4	the	the	DET
asir-36063	258	5	second	second	ADJ
asir-36063	258	6	smallest	small	ADJ
asir-36063	258	7	eigenvalue	eigenvalue	NOUN
asir-36063	258	8	of	of	ADP
asir-36063	258	9	𝚲	𝚲	NOUN
asir-36063	258	10	is	be	AUX
asir-36063	258	11	non	non	ADJ
asir-36063	258	12	-	-	ADJ
asir-36063	258	13	zero	zero	NUM
asir-36063	258	14	,	,	PUNCT
asir-36063	258	15	together	together	ADV
asir-36063	258	16	with	with	ADP
asir-36063	258	17	the	the	DET
asir-36063	258	18	complementary	complementary	ADJ
asir-36063	258	19	slackness	slackness	NOUN
asir-36063	258	20	,	,	PUNCT
asir-36063	258	21	the	the	DET
asir-36063	258	22	fact	fact	NOUN
asir-36063	258	23	that	that	SCONJ
asir-36063	258	24	𝑿′	𝑿′	PROPN
asir-36063	258	25	≽	≽	NOUN
asir-36063	258	26	0	0	PUNCT
asir-36063	258	27	and	and	CCONJ
asir-36063	258	28	𝚲	𝚲	NOUN
asir-36063	258	29	≽	≽	NOUN
asir-36063	258	30	0	0	NUM
asir-36063	258	31	,	,	PUNCT
asir-36063	258	32	we	we	PRON
asir-36063	258	33	have	have	AUX
asir-36063	258	34	𝑿′	𝑿′	PROPN
asir-36063	259	1	=	=	SYM
asir-36063	259	2	𝑘𝒈𝒈⊤	𝑘𝒈𝒈⊤	PROPN
asir-36063	259	3	.	.	PUNCT
asir-36063	260	1	since	since	SCONJ
asir-36063	260	2	𝑿𝑖𝑖	𝑿𝑖𝑖	PROPN
asir-36063	260	3	′	′	NUM
asir-36063	260	4	=	=	SYM
asir-36063	260	5	1,𝑿′	1,𝑿′	NUM
asir-36063	260	6	=	=	SYM
asir-36063	260	7	𝒈𝒈⊤	𝒈𝒈⊤	ADV
asir-36063	260	8	by	by	ADP
asir-36063	260	9	contradiction	contradiction	NOUN
asir-36063	260	10	.	.	PUNCT
asir-36063	261	1	then	then	ADV
asir-36063	261	2	we	we	PRON
asir-36063	261	3	need	need	VERB
asir-36063	261	4	to	to	PART
asir-36063	261	5	estimate	estimate	VERB
asir-36063	261	6	𝔼𝚲	𝔼𝚲	PROPN
asir-36063	261	7	,	,	PUNCT
asir-36063	261	8	then	then	ADV
asir-36063	261	9	we	we	PRON
asir-36063	261	10	have	have	VERB
asir-36063	261	11	the	the	DET
asir-36063	261	12	following	following	NOUN
asir-36063	261	13	:	:	PUNCT
asir-36063	261	14	𝔼[𝚲]⁡=	𝔼[𝚲]⁡=	PROPN
asir-36063	261	15	𝔼	𝔼	PROPN
asir-36063	261	16	[	[	X
asir-36063	261	17	2γ𝑆𝐵𝑀	2γ𝑆𝐵𝑀	NOUN
asir-36063	261	18	+	+	X
asir-36063	261	19	𝑰2𝑛	𝑰2𝑛	X
asir-36063	262	1	+	+	PUNCT
asir-36063	262	2	[	[	PUNCT
asir-36063	262	3	0	0	X
asir-36063	262	4	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	262	5	−	−	NUM
asir-36063	262	6	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	262	7	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	262	8	−	−	PROPN
asir-36063	262	9	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	262	10	0	0	NUM
asir-36063	262	11	]	]	PUNCT
asir-36063	263	1	+	+	CCONJ
asir-36063	263	2	2	2	NUM
asir-36063	263	3	[	[	PUNCT
asir-36063	263	4	𝑱𝑛	𝑱𝑛	NOUN
asir-36063	263	5	0	0	NUM
asir-36063	263	6	0	0	NUM
asir-36063	263	7	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	263	8	]	]	PUNCT
asir-36063	263	9	]	]	X
asir-36063	263	10	⁡=	⁡=	NOUN
asir-36063	263	11	2	2	NUM
asir-36063	263	12	(	(	PUNCT
asir-36063	263	13	𝑛	𝑛	PROPN
asir-36063	263	14	2	2	NUM
asir-36063	263	15	(	(	PUNCT
asir-36063	263	16	𝑝	𝑝	NOUN
asir-36063	263	17	−	−	PROPN
asir-36063	263	18	𝑞)𝑰2𝑛	𝑞)𝑰2𝑛	X
asir-36063	263	19	−	−	PROPN
asir-36063	263	20	(	(	PUNCT
asir-36063	263	21	𝑝+𝑞	𝑝+𝑞	NOUN
asir-36063	263	22	2	2	NUM
asir-36063	263	23	[	[	PUNCT
asir-36063	263	24	0	0	NUM
asir-36063	263	25	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	263	26	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	263	27	0	0	NUM
asir-36063	263	28	]	]	PUNCT
asir-36063	264	1	+	+	CCONJ
asir-36063	264	2	𝑝−𝑞	𝑝−𝑞	ADJ
asir-36063	264	3	2	2	NUM
asir-36063	264	4	𝒈𝒈⊤	𝒈𝒈⊤	NUM
asir-36063	264	5	)	)	PUNCT
asir-36063	264	6	)	)	PUNCT
asir-36063	265	1	⁡+	⁡+	PUNCT
asir-36063	265	2	[	[	PUNCT
asir-36063	265	3	0	0	NUM
asir-36063	265	4	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	265	5	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	265	6	0	0	NUM
asir-36063	265	7	]	]	PUNCT
asir-36063	266	1	+	+	CCONJ
asir-36063	266	2	𝑰2𝑛	𝑰2𝑛	PROPN
asir-36063	266	3	−	−	PROPN
asir-36063	266	4	[	[	PUNCT
asir-36063	266	5	0	0	NUM
asir-36063	266	6	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	266	7	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	266	8	0	0	NUM
asir-36063	266	9	]	]	PUNCT
asir-36063	267	1	+	+	CCONJ
asir-36063	267	2	(	(	PUNCT
asir-36063	267	3	𝑝	𝑝	PROPN
asir-36063	267	4	−	−	PROPN
asir-36063	267	5	𝑞	𝑞	NOUN
asir-36063	267	6	)	)	PUNCT
asir-36063	267	7	[	[	PUNCT
asir-36063	267	8	𝒈′𝒈′⊤	𝒈′𝒈′⊤	X
asir-36063	267	9	0	0	NUM
asir-36063	267	10	0	0	NUM
asir-36063	267	11	𝒈′𝒈′⊤	𝒈′𝒈′⊤	NOUN
asir-36063	267	12	]	]	X
asir-36063	268	1	+	+	NUM
asir-36063	268	2	2	2	NUM
asir-36063	268	3	[	[	PUNCT
asir-36063	268	4	𝑱𝑛	𝑱𝑛	NOUN
asir-36063	268	5	0	0	NUM
asir-36063	268	6	0	0	NUM
asir-36063	268	7	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	268	8	]	]	PUNCT
asir-36063	268	9	⁡=	⁡=	NOUN
asir-36063	269	1	𝑛(𝑝	𝑛(𝑝	ADJ
asir-36063	269	2	−	−	X
asir-36063	270	1	𝑞)(𝑰2𝑛	𝑞)(𝑰2𝑛	X
asir-36063	270	2	−	−	PROPN
asir-36063	270	3	[	[	PUNCT
asir-36063	270	4	0	0	NUM
asir-36063	270	5	𝒈′𝒈′⊤	𝒈′𝒈′⊤	X
asir-36063	270	6	𝒈′𝒈′⊤	𝒈′𝒈′⊤	NOUN
asir-36063	270	7	0	0	PUNCT
asir-36063	270	8	]	]	SYM
asir-36063	270	9	𝑛	𝑛	X
asir-36063	270	10	)	)	PUNCT
asir-36063	271	1	+	+	CCONJ
asir-36063	271	2	(	(	PUNCT
asir-36063	271	3	1	1	NUM
asir-36063	271	4	−	−	PROPN
asir-36063	271	5	(	(	PUNCT
asir-36063	271	6	𝑝	𝑝	PROPN
asir-36063	271	7	+	+	NUM
asir-36063	271	8	𝑞	𝑞	NOUN
asir-36063	271	9	)	)	PUNCT
asir-36063	271	10	)	)	PUNCT
asir-36063	272	1	[	[	PUNCT
asir-36063	272	2	0	0	NUM
asir-36063	272	3	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	272	4	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	272	5	0	0	NUM
asir-36063	272	6	]	]	PUNCT
asir-36063	273	1	+	+	CCONJ
asir-36063	273	2	𝑰2𝑛	𝑰2𝑛	PROPN
asir-36063	273	3	−	−	PROPN
asir-36063	273	4	[	[	PUNCT
asir-36063	273	5	0	0	NUM
asir-36063	273	6	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	273	7	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	273	8	0	0	NUM
asir-36063	273	9	]	]	PUNCT
asir-36063	274	1	+	+	CCONJ
asir-36063	274	2	2	2	NUM
asir-36063	274	3	[	[	PUNCT
asir-36063	274	4	𝑱𝑛	𝑱𝑛	NOUN
asir-36063	274	5	0	0	NUM
asir-36063	274	6	0	0	NUM
asir-36063	274	7	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	274	8	]	]	PUNCT
asir-36063	274	9	(	(	PUNCT
asir-36063	274	10	b.3	b.3	X
asir-36063	274	11	)	)	PUNCT
asir-36063	274	12	suppose	suppose	VERB
asir-36063	274	13	𝑝	𝑝	ADP
asir-36063	274	14	<	<	X
asir-36063	274	15	1	1	NUM
asir-36063	274	16	2	2	NUM
asir-36063	274	17	and	and	CCONJ
asir-36063	274	18	𝜆2	𝜆2	NOUN
asir-36063	274	19	=	=	SYM
asir-36063	275	1	𝑛(𝑝	𝑛(𝑝	PROPN
asir-36063	275	2	−	−	PUNCT
asir-36063	275	3	𝑞	𝑞	NOUN
asir-36063	275	4	)	)	PUNCT
asir-36063	275	5	,	,	PUNCT
asir-36063	275	6	whose	whose	DET
asir-36063	275	7	eigenvector	eigenvector	NOUN
asir-36063	275	8	is	be	AUX
asir-36063	275	9	perpendicular	perpendicular	ADJ
asir-36063	275	10	to	to	ADP
asir-36063	275	11	(	(	PUNCT
asir-36063	275	12	𝒈′	𝒈′	PROPN
asir-36063	275	13	,	,	PUNCT
asir-36063	275	14	𝒈′)⊤	𝒈′)⊤	X
asir-36063	275	15	and	and	CCONJ
asir-36063	275	16	(	(	PUNCT
asir-36063	275	17	𝟏	𝟏	NUM
asir-36063	275	18	,	,	PUNCT
asir-36063	275	19	𝟏)⊤𝚫	𝟏)⊤𝚫	PROPN
asir-36063	275	20	can	can	AUX
asir-36063	275	21	be	be	AUX
asir-36063	275	22	re	re	VERB
asir-36063	275	23	-	-	VERB
asir-36063	275	24	written	write	VERB
asir-36063	275	25	as	as	ADP
asir-36063	275	26	:	:	PUNCT
asir-36063	275	27	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝚲	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝚲	ADV
asir-36063	275	28	=	=	PUNCT
asir-36063	275	29	2γdsbm	2γdsbm	NUM
asir-36063	275	30	+	+	CCONJ
asir-36063	275	31	𝑰2𝑛	𝑰2𝑛	PROPN
asir-36063	276	1	+	+	CCONJ
asir-36063	276	2	[	[	PUNCT
asir-36063	276	3	0	0	X
asir-36063	276	4	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	276	5	−	−	NUM
asir-36063	276	6	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	276	7	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	276	8	−	−	PROPN
asir-36063	276	9	𝑰𝑛	𝑰𝑛	PROPN
asir-36063	276	10	0	0	NUM
asir-36063	276	11	]	]	PUNCT
asir-36063	277	1	+	+	CCONJ
asir-36063	277	2	2	2	NUM
asir-36063	277	3	[	[	PUNCT
asir-36063	277	4	𝑱𝑛	𝑱𝑛	NOUN
asir-36063	277	5	0	0	NUM
asir-36063	277	6	0	0	NUM
asir-36063	277	7	𝑱𝑛	𝑱𝑛	PROPN
asir-36063	277	8	]	]	PUNCT
asir-36063	277	9	(	(	PUNCT
asir-36063	277	10	b.4	b.4	NOUN
asir-36063	277	11	)	)	PUNCT
asir-36063	277	12	using	use	VERB
asir-36063	277	13	weyl	weyl	PROPN
asir-36063	277	14	's	's	PART
asir-36063	277	15	inequalities	inequality	NOUN
asir-36063	277	16	,	,	PUNCT
asir-36063	277	17	we	we	PRON
asir-36063	277	18	get	get	VERB
asir-36063	277	19	:	:	PUNCT
asir-36063	277	20	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝜆2	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝜆2	NOUN
asir-36063	277	21	>	>	X
asir-36063	277	22	𝜆max(𝔼[𝚲	𝜆max(𝔼[𝚲	PROPN
asir-36063	277	23	]	]	X
asir-36063	277	24	−	−	PROPN
asir-36063	277	25	𝚲	𝚲	NOUN
asir-36063	277	26	)	)	PUNCT
asir-36063	277	27	=	=	NOUN
asir-36063	277	28	∥	∥	X
asir-36063	277	29	𝔼[𝚲	𝔼[𝚲	NOUN
asir-36063	277	30	]	]	PUNCT
asir-36063	278	1	−	−	PUNCT
asir-36063	278	2	𝚲	𝚲	NOUN
asir-36063	278	3	∥	∥	X
asir-36063	278	4	⁡≥	⁡≥	PUNCT
asir-36063	278	5	𝜎2(𝔼[𝚲	𝜎2(𝔼[𝚲	NUM
asir-36063	278	6	]	]	X
asir-36063	278	7	−	−	PROPN
asir-36063	278	8	𝜎2(𝚲	𝜎2(𝚲	NUM
asir-36063	278	9	)	)	PUNCT
asir-36063	278	10	)	)	PUNCT
asir-36063	278	11	⁡=	⁡=	NOUN
asir-36063	278	12	𝜆2(𝔼[𝚲	𝜆2(𝔼[𝚲	PROPN
asir-36063	278	13	]	]	PUNCT
asir-36063	278	14	−	−	PROPN
asir-36063	278	15	𝜆2(𝚲	𝜆2(𝚲	NOUN
asir-36063	278	16	)	)	PUNCT
asir-36063	278	17	)	)	PUNCT
asir-36063	278	18	⁡≥	⁡≥	PUNCT
asir-36063	278	19	𝜆2(𝔼[𝚲	𝜆2(𝔼[𝚲	PUNCT
asir-36063	278	20	]	]	PUNCT
asir-36063	278	21	−	−	PROPN
asir-36063	278	22	𝜆2(𝚲	𝜆2(𝚲	NOUN
asir-36063	278	23	)	)	PUNCT
asir-36063	278	24	)	)	PUNCT
asir-36063	278	25	(	(	PUNCT
asir-36063	278	26	b.5	b.5	NOUN
asir-36063	278	27	)	)	PUNCT
asir-36063	278	28	this	this	PRON
asir-36063	278	29	implies	imply	VERB
asir-36063	278	30	that	that	SCONJ
asir-36063	278	31	𝒈𝒈⊤	𝒈𝒈⊤	NOUN
asir-36063	278	32	is	be	AUX
asir-36063	278	33	the	the	DET
asir-36063	278	34	unique	unique	ADJ
asir-36063	278	35	solution	solution	NOUN
asir-36063	278	36	to	to	ADP
asir-36063	278	37	the	the	DET
asir-36063	278	38	semidefinite	semidefinite	PROPN
asir-36063	278	39	programming	programming	NOUN
asir-36063	278	40	.	.	PUNCT
asir-36063	279	1	proof	proof	NOUN
asir-36063	279	2	for	for	ADP
asir-36063	279	3	theorem	theorem	ADJ
asir-36063	279	4	3.6	3.6	NUM
asir-36063	279	5	the	the	DET
asir-36063	279	6	method	method	NOUN
asir-36063	279	7	to	to	PART
asir-36063	279	8	approach	approach	VERB
asir-36063	279	9	the	the	DET
asir-36063	279	10	problem	problem	NOUN
asir-36063	279	11	is	be	AUX
asir-36063	279	12	by	by	ADP
asir-36063	279	13	applying	apply	VERB
asir-36063	279	14	spectral	spectral	ADJ
asir-36063	279	15	approximation	approximation	NOUN
asir-36063	279	16	of	of	ADP
asir-36063	279	17	random	random	ADJ
asir-36063	279	18	laplacian	laplacian	ADJ
asir-36063	279	19	matrix	matrix	NOUN
asir-36063	279	20	algorithm	algorithm	NOUN
asir-36063	279	21	.	.	PUNCT
asir-36063	280	1	however	however	ADV
asir-36063	280	2	,	,	PUNCT
asir-36063	280	3	γdsbm	γdsbm	NOUN
asir-36063	280	4	is	be	AUX
asir-36063	280	5	not	not	PART
asir-36063	280	6	a	a	DET
asir-36063	280	7	laplacian	laplacian	ADJ
asir-36063	280	8	matrix	matrix	NOUN
asir-36063	280	9	.	.	PUNCT
asir-36063	281	1	therefore	therefore	ADV
asir-36063	281	2	,	,	PUNCT
asir-36063	281	3	we	we	PRON
asir-36063	281	4	will	will	AUX
asir-36063	281	5	try	try	VERB
asir-36063	281	6	to	to	PART
asir-36063	281	7	construct	construct	VERB
asir-36063	281	8	a	a	DET
asir-36063	281	9	laplacian	laplacian	ADJ
asir-36063	281	10	matrix	matrix	NOUN
asir-36063	281	11	γdsbm	γdsbm	NOUN
asir-36063	281	12	′	′	NUM
asir-36063	281	13	to	to	PART
asir-36063	281	14	help	help	VERB
asir-36063	281	15	solve	solve	VERB
asir-36063	281	16	the	the	DET
asir-36063	281	17	problem	problem	NOUN
asir-36063	281	18	.	.	PUNCT
asir-36063	282	1	w.l.o.g	w.l.o.g	PROPN
asir-36063	282	2	.	.	PUNCT
asir-36063	283	1	we	we	PRON
asir-36063	283	2	let	let	VERB
asir-36063	283	3	𝒈	𝒈	PRON
asir-36063	283	4	=	=	PUNCT
asir-36063	283	5	(	(	PUNCT
asir-36063	283	6	𝟏𝑛/2	𝟏𝑛/2	PROPN
asir-36063	283	7	,	,	PUNCT
asir-36063	283	8	−𝟏𝑛/2	−𝟏𝑛/2	PRON
asir-36063	283	9	,	,	PUNCT
asir-36063	283	10	𝟏𝑛/2	𝟏𝑛/2	PROPN
asir-36063	283	11	,	,	PUNCT
asir-36063	283	12	−𝟏𝑛/2	−𝟏𝑛/2	NOUN
asir-36063	283	13	)	)	PUNCT
asir-36063	283	14	,	,	PUNCT
asir-36063	283	15	and	and	CCONJ
asir-36063	283	16	we	we	PRON
asir-36063	283	17	define	define	VERB
asir-36063	283	18	:	:	PUNCT
asir-36063	283	19	γdsbm	γdsbm	ADJ
asir-36063	283	20	′	′	NUM
asir-36063	283	21	=	=	SYM
asir-36063	283	22	diag⁡(𝒈)γdsbmdiag⁡(𝒈	diag⁡(𝒈)γdsbmdiag⁡(𝒈	NOUN
asir-36063	283	23	)	)	PUNCT
asir-36063	283	24	(	(	PUNCT
asir-36063	283	25	b.6	b.6	ADP
asir-36063	283	26	)	)	PUNCT
asir-36063	283	27	both	both	CCONJ
asir-36063	283	28	the	the	DET
asir-36063	283	29	eigenvalue	eigenvalue	PROPN
asir-36063	283	30	and	and	CCONJ
asir-36063	283	31	the	the	DET
asir-36063	283	32	diagonal	diagonal	ADJ
asir-36063	283	33	elements	element	NOUN
asir-36063	283	34	of	of	ADP
asir-36063	283	35	𝔼[γdsbm	𝔼[γdsbm	PROPN
asir-36063	283	36	′	′	NUM
asir-36063	283	37	]	]	PUNCT
asir-36063	284	1	−	−	PUNCT
asir-36063	284	2	γdsbm	γdsbm	NOUN
asir-36063	284	3	′	′	NUM
asir-36063	284	4	are	be	AUX
asir-36063	284	5	the	the	DET
asir-36063	284	6	same	same	ADJ
asir-36063	284	7	as	as	ADP
asir-36063	284	8	those	those	PRON
asir-36063	284	9	of	of	ADP
asir-36063	284	10	𝔼[γdsbm	𝔼[γdsbm	NOUN
asir-36063	284	11	]	]	X
asir-36063	284	12	−	−	PUNCT
asir-36063	284	13	γdsbm	γdsbm	NOUN
asir-36063	284	14	.	.	PUNCT
asir-36063	285	1	the	the	DET
asir-36063	285	2	off	off	ADV
asir-36063	285	3	-	-	PUNCT
asir-36063	285	4	diagonal	diagonal	ADJ
asir-36063	285	5	entries	entry	NOUN
asir-36063	285	6	of	of	ADP
asir-36063	285	7	γdsbm	γdsbm	NOUN
asir-36063	285	8	′	′	NUM
asir-36063	285	9	=	=	SYM
asir-36063	285	10	−𝑨𝑖𝑗𝑔𝑖𝑔𝑖	−𝑨𝑖𝑗𝑔𝑖𝑔𝑖	PROPN
asir-36063	285	11	.	.	PUNCT
asir-36063	286	1	then	then	ADV
asir-36063	286	2	we	we	PRON
asir-36063	286	3	apply	apply	VERB
asir-36063	286	4	spectral	spectral	ADJ
asir-36063	286	5	approximation	approximation	NOUN
asir-36063	286	6	of	of	ADP
asir-36063	286	7	random	random	ADJ
asir-36063	286	8	laplacian	laplacian	ADJ
asir-36063	286	9	matrix	matrix	NOUN
asir-36063	286	10	algorithm	algorithm	NOUN
asir-36063	286	11	,	,	PUNCT
asir-36063	286	12	we	we	PRON
asir-36063	286	13	let	let	VERB
asir-36063	286	14	𝑳	𝑳	NOUN
asir-36063	286	15	=	=	PUNCT
asir-36063	286	16	𝔼[γdsbm	𝔼[γdsbm	PROPN
asir-36063	286	17	′	′	NOUN
asir-36063	286	18	]	]	PUNCT
asir-36063	287	1	−	−	PUNCT
asir-36063	287	2	γdsbm	γdsbm	NOUN
asir-36063	287	3	′	′	NUM
asir-36063	287	4	,	,	PUNCT
asir-36063	287	5	where	where	SCONJ
asir-36063	287	6	𝑳	𝑳	NOUN
asir-36063	287	7	has	have	VERB
asir-36063	287	8	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	287	9	applied	apply	VERB
asir-36063	287	10	science	science	NOUN
asir-36063	287	11	and	and	CCONJ
asir-36063	287	12	innovative	innovative	ADJ
asir-36063	287	13	research	research	NOUN
asir-36063	287	14	vol	vol	NOUN
asir-36063	287	15	.	.	PROPN
asir-36063	287	16	8	8	NUM
asir-36063	287	17	,	,	PUNCT
asir-36063	287	18	no	no	INTJ
asir-36063	287	19	.	.	NOUN
asir-36063	287	20	1	1	NUM
asir-36063	287	21	,	,	PUNCT
asir-36063	287	22	2024	2024	NUM
asir-36063	287	23	75	75	NUM
asir-36063	287	24	published	publish	VERB
asir-36063	287	25	by	by	ADP
asir-36063	287	26	scholink	scholink	PROPN
asir-36063	287	27	inc	inc	PROPN
asir-36063	287	28	.	.	PROPN
asir-36063	287	29	independent	independent	ADJ
asir-36063	287	30	off	off	ADP
asir-36063	287	31	-	-	PUNCT
asir-36063	287	32	diagonal	diagonal	ADJ
asir-36063	287	33	entries	entry	NOUN
asir-36063	287	34	.	.	PUNCT
asir-36063	288	1	then	then	ADV
asir-36063	288	2	we	we	PRON
asir-36063	288	3	have	have	VERB
asir-36063	288	4	:	:	PUNCT
asir-36063	288	5	∑	∑	PART
asir-36063	288	6	 	 	SPACE
asir-36063	288	7	𝑗∈[2𝑛]/𝑖	𝑗∈[2𝑛]/𝑖	PROPN
asir-36063	288	8	𝔼[𝑳𝑖𝑗	𝔼[𝑳𝑖𝑗	PROPN
asir-36063	288	9	2	2	NUM
asir-36063	288	10	]	]	PUNCT
asir-36063	288	11	=	=	SYM
asir-36063	288	12	(	(	PUNCT
asir-36063	288	13	𝑛	𝑛	PROPN
asir-36063	288	14	2	2	NUM
asir-36063	288	15	−	−	NOUN
asir-36063	288	16	1)𝑝(1	1)𝑝(1	NUM
asir-36063	288	17	−	−	PROPN
asir-36063	288	18	𝑝	𝑝	NOUN
asir-36063	288	19	)	)	PUNCT
asir-36063	289	1	+	+	CCONJ
asir-36063	289	2	𝑛	𝑛	DET
asir-36063	289	3	2	2	NUM
asir-36063	289	4	𝑞(1	𝑞(1	PROPN
asir-36063	289	5	−	−	NUM
asir-36063	289	6	𝑞	𝑞	NOUN
asir-36063	289	7	)	)	PUNCT
asir-36063	289	8	≤	≤	NUM
asir-36063	289	9	𝑛	𝑛	DET
asir-36063	289	10	2	2	NUM
asir-36063	289	11	∗	∗	NOUN
asir-36063	289	12	1	1	NUM
asir-36063	289	13	4	4	NUM
asir-36063	289	14	(	(	PUNCT
asir-36063	289	15	𝑝	𝑝	PROPN
asir-36063	289	16	+	+	NUM
asir-36063	289	17	𝑞	𝑞	NOUN
asir-36063	289	18	)	)	PUNCT
asir-36063	289	19	>	>	X
asir-36063	290	1	𝑛	𝑛	PRON
asir-36063	290	2	8	8	NUM
asir-36063	290	3	2log⁡𝑛	2log⁡𝑛	NUM
asir-36063	290	4	𝑐𝑛	𝑐𝑛	PROPN
asir-36063	290	5	(	(	PUNCT
asir-36063	290	6	1	1	NUM
asir-36063	290	7	−	−	NOUN
asir-36063	290	8	𝑞)2	𝑞)2	NOUN
asir-36063	290	9	=	=	SYM
asir-36063	290	10	log⁡𝑛	log⁡𝑛	NOUN
asir-36063	290	11	4𝑐	4𝑐	PROPN
asir-36063	290	12	max	max	PROPN
asir-36063	290	13	𝑖≠𝑗	𝑖≠𝑗	PROPN
asir-36063	290	14	 	 	SPACE
asir-36063	290	15	∥∥𝑳𝑖𝑗∥∥∞	∥∥𝑳𝑖𝑗∥∥∞	X
asir-36063	290	16	2	2	NUM
asir-36063	290	17	(	(	PUNCT
asir-36063	290	18	b.7	b.7	NOUN
asir-36063	290	19	)	)	PUNCT
asir-36063	290	20	where	where	SCONJ
asir-36063	290	21	it	it	PRON
asir-36063	290	22	exists	exist	VERB
asir-36063	290	23	a	a	DET
asir-36063	290	24	constant	constant	ADJ
asir-36063	290	25	𝚫′	𝚫′	NOUN
asir-36063	290	26	such	such	ADJ
asir-36063	290	27	that	that	PRON
asir-36063	290	28	with	with	ADP
asir-36063	290	29	high	high	ADJ
asir-36063	290	30	probability	probability	NOUN
asir-36063	290	31	,	,	PUNCT
asir-36063	290	32	𝜆max(𝔼[γdsbm	𝜆max(𝔼[γdsbm	PROPN
asir-36063	290	33	′	′	NUM
asir-36063	290	34	]	]	PUNCT
asir-36063	291	1	−	−	NUM
asir-36063	291	2	γ𝐷𝑆𝐵𝑀	γ𝐷𝑆𝐵𝑀	NOUN
asir-36063	291	3	′	′	NUM
asir-36063	291	4	)	)	PUNCT
asir-36063	291	5	≤	≤	NOUN
asir-36063	291	6	(	(	PUNCT
asir-36063	291	7	1	1	NUM
asir-36063	291	8	+	+	CCONJ
asir-36063	291	9	𝚫′	𝚫′	PROPN
asir-36063	291	10	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	291	11	)	)	PUNCT
asir-36063	291	12	max	max	PROPN
asir-36063	291	13	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	291	14	]	]	PUNCT
asir-36063	291	15	 	 	SPACE
asir-36063	292	1	[	[	X
asir-36063	292	2	𝔼[(γdsbm	𝔼[(γdsbm	PROPN
asir-36063	292	3	′	′	NUM
asir-36063	292	4	)	)	PUNCT
asir-36063	292	5	𝑖𝑖	𝑖𝑖	ADP
asir-36063	292	6	]	]	X
asir-36063	292	7	−	−	PROPN
asir-36063	292	8	(	(	PUNCT
asir-36063	292	9	γdsbm	γdsbm	PROPN
asir-36063	292	10	′	′	NUM
asir-36063	292	11	)	)	PUNCT
asir-36063	292	12	𝑖𝑖	𝑖𝑖	X
asir-36063	292	13	]	]	X
asir-36063	292	14	(	(	PUNCT
asir-36063	292	15	b.8	b.8	NUM
asir-36063	292	16	)	)	PUNCT
asir-36063	292	17	and	and	CCONJ
asir-36063	292	18	it	it	PRON
asir-36063	292	19	is	be	AUX
asir-36063	292	20	the	the	DET
asir-36063	292	21	same	same	ADJ
asir-36063	292	22	as	as	ADP
asir-36063	292	23	the	the	DET
asir-36063	292	24	following	following	NOUN
asir-36063	292	25	:	:	PUNCT
asir-36063	292	26	𝜆max(𝔼[γdsbm	𝜆max(𝔼[γdsbm	X
asir-36063	292	27	]	]	X
asir-36063	293	1	−	−	NUM
asir-36063	293	2	γ𝐷𝑆𝐵𝑀	γ𝐷𝑆𝐵𝑀	NOUN
asir-36063	293	3	)	)	PUNCT
asir-36063	293	4	≤	≤	NOUN
asir-36063	293	5	(	(	PUNCT
asir-36063	293	6	1	1	NUM
asir-36063	293	7	+	+	CCONJ
asir-36063	293	8	𝚫′	𝚫′	PROPN
asir-36063	293	9	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	293	10	)	)	PUNCT
asir-36063	293	11	max	max	PROPN
asir-36063	293	12	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	293	13	]	]	PUNCT
asir-36063	293	14	 	 	SPACE
asir-36063	294	1	[	[	X
asir-36063	294	2	𝔼[(γdsbm)𝑖𝑖	𝔼[(γdsbm)𝑖𝑖	X
asir-36063	294	3	]	]	X
asir-36063	294	4	−	−	PROPN
asir-36063	294	5	(	(	PUNCT
asir-36063	294	6	γdsbm)𝑖𝑖	γdsbm)𝑖𝑖	PROPN
asir-36063	294	7	]	]	X
asir-36063	294	8	(	(	PUNCT
asir-36063	294	9	b.9	b.9	X
asir-36063	294	10	)	)	PUNCT
asir-36063	294	11	it	it	PRON
asir-36063	294	12	is	be	AUX
asir-36063	294	13	worth	worth	ADJ
asir-36063	294	14	mentioning	mention	VERB
asir-36063	294	15	that	that	SCONJ
asir-36063	294	16	⁡⁡⁡⁡⁡⁡	⁡⁡⁡⁡⁡⁡	ADJ
asir-36063	294	17	min	min	NOUN
asir-36063	294	18	𝑖∈[2𝑛	𝑖∈[2𝑛	X
asir-36063	294	19	]	]	PUNCT
asir-36063	294	20	 	 	SPACE
asir-36063	294	21	(	(	PUNCT
asir-36063	294	22	(	(	PUNCT
asir-36063	294	23	𝒟+)𝑖𝑖	𝒟+)𝑖𝑖	INTJ
asir-36063	294	24	−	−	NOUN
asir-36063	295	1	(	(	PUNCT
asir-36063	295	2	𝒟−)𝑖𝑖)⁡≥	𝒟−)𝑖𝑖)⁡≥	PROPN
asir-36063	295	3	(	(	PUNCT
asir-36063	295	4	1	1	NUM
asir-36063	295	5	+	+	CCONJ
asir-36063	295	6	𝚫′	𝚫′	PROPN
asir-36063	295	7	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	295	8	)	)	PUNCT
asir-36063	295	9	max	max	PROPN
asir-36063	295	10	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	295	11	]	]	PUNCT
asir-36063	295	12	 	 	SPACE
asir-36063	296	1	[	[	X
asir-36063	296	2	𝔼[(γdsbm)𝑖𝑖	𝔼[(γdsbm)𝑖𝑖	X
asir-36063	296	3	]	]	X
asir-36063	296	4	−	−	PROPN
asir-36063	296	5	(	(	PUNCT
asir-36063	296	6	γdsbm)𝑖𝑖	γdsbm)𝑖𝑖	PROPN
asir-36063	296	7	]	]	X
asir-36063	296	8	⁡=	⁡=	NOUN
asir-36063	296	9	(	(	PUNCT
asir-36063	296	10	1	1	NUM
asir-36063	296	11	−	−	NOUN
asir-36063	296	12	𝚫′	𝚫′	PROPN
asir-36063	296	13	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	296	14	)	)	PUNCT
asir-36063	296	15	(	(	PUNCT
asir-36063	296	16	𝑛	𝑛	PROPN
asir-36063	296	17	2	2	NUM
asir-36063	296	18	(	(	PUNCT
asir-36063	296	19	𝑝	𝑝	PROPN
asir-36063	296	20	−	−	PROPN
asir-36063	296	21	𝑞	𝑞	NOUN
asir-36063	296	22	)	)	PUNCT
asir-36063	296	23	−	−	PROPN
asir-36063	297	1	𝑝)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	𝑝)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	297	2	.	.	NOUN
asir-36063	297	3	10	10	NUM
asir-36063	297	4	)	)	PUNCT
asir-36063	297	5	⁡≥	⁡≥	NUM
asir-36063	297	6	max	max	X
asir-36063	297	7	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	297	8	]	]	PUNCT
asir-36063	297	9	 	 	SPACE
asir-36063	297	10	(	(	PUNCT
asir-36063	297	11	𝔼[(γdsbm	𝔼[(γdsbm	PROPN
asir-36063	297	12	′	′	NUM
asir-36063	297	13	)	)	PUNCT
asir-36063	297	14	𝑖𝑖	𝑖𝑖	ADP
asir-36063	297	15	]	]	X
asir-36063	297	16	−	−	PROPN
asir-36063	297	17	(	(	PUNCT
asir-36063	297	18	γdsbm	γdsbm	PROPN
asir-36063	297	19	′	′	NUM
asir-36063	297	20	)	)	PUNCT
asir-36063	297	21	𝑖𝑖	𝑖𝑖	PUNCT
asir-36063	297	22	therefore	therefore	ADV
asir-36063	297	23	we	we	PRON
asir-36063	297	24	have	have	VERB
asir-36063	297	25	:	:	PUNCT
asir-36063	297	26	𝜆max(𝔼[γdsbm	𝜆max(𝔼[γdsbm	NOUN
asir-36063	298	1	′	′	NOUN
asir-36063	298	2	]	]	PUNCT
asir-36063	299	1	−	−	NUM
asir-36063	299	2	γ𝐷𝑆𝐵𝑀	γ𝐷𝑆𝐵𝑀	NOUN
asir-36063	299	3	′	′	NUM
asir-36063	299	4	)	)	PUNCT
asir-36063	299	5	≤	≤	NOUN
asir-36063	299	6	(	(	PUNCT
asir-36063	299	7	1	1	NUM
asir-36063	299	8	+	+	CCONJ
asir-36063	299	9	𝚫′	𝚫′	PROPN
asir-36063	299	10	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	299	11	)	)	PUNCT
asir-36063	299	12	(	(	PUNCT
asir-36063	299	13	1	1	NUM
asir-36063	299	14	−	−	NOUN
asir-36063	299	15	𝚫′	𝚫′	PROPN
asir-36063	299	16	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	299	17	)	)	PUNCT
asir-36063	299	18	(	(	PUNCT
asir-36063	299	19	𝑛	𝑛	PROPN
asir-36063	299	20	2	2	NUM
asir-36063	299	21	(	(	PUNCT
asir-36063	299	22	𝑝	𝑝	PROPN
asir-36063	299	23	−	−	PROPN
asir-36063	299	24	𝑞	𝑞	PROPN
asir-36063	299	25	)	)	PUNCT
asir-36063	299	26	−	−	PROPN
asir-36063	299	27	𝑝)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	𝑝)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	299	28	.	.	PUNCT
asir-36063	300	1	11	11	NUM
asir-36063	300	2	)	)	PUNCT
asir-36063	300	3	for	for	ADP
asir-36063	300	4	each	each	DET
asir-36063	300	5	𝚫′	𝚫′	NOUN
asir-36063	300	6	,	,	PUNCT
asir-36063	300	7	there	there	PRON
asir-36063	300	8	exists	exist	VERB
asir-36063	300	9	at	at	ADP
asir-36063	300	10	least	least	ADV
asir-36063	300	11	one	one	NUM
asir-36063	300	12	𝚫′	𝚫′	NOUN
asir-36063	300	13	>	>	X
asir-36063	300	14	0	0	NUM
asir-36063	301	1	such	such	ADJ
asir-36063	301	2	that	that	SCONJ
asir-36063	301	3	:	:	PUNCT
asir-36063	301	4	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(1	PROPN
asir-36063	301	5	−	−	PROPN
asir-36063	301	6	𝚫′	𝚫′	PROPN
asir-36063	301	7	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	301	8	)	)	PUNCT
asir-36063	301	9	(	(	PUNCT
asir-36063	301	10	1	1	NUM
asir-36063	301	11	+	+	NUM
asir-36063	301	12	𝚫′	𝚫′	PROPN
asir-36063	301	13	√log⁡𝑛	√log⁡𝑛	NOUN
asir-36063	301	14	)	)	PUNCT
asir-36063	301	15	<	<	X
asir-36063	302	1	1⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	1⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	PROPN
asir-36063	302	2	.	.	PUNCT
asir-36063	302	3	12	12	NUM
asir-36063	302	4	)	)	PUNCT
asir-36063	303	1	hence	hence	ADV
asir-36063	303	2	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝜆max(𝔼[γdsbm	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝜆max(𝔼[γdsbm	ADP
asir-36063	303	3	′	′	ADP
asir-36063	303	4	]	]	PUNCT
asir-36063	304	1	−	−	NUM
asir-36063	304	2	γ𝐷𝑆𝐵𝑀	γ𝐷𝑆𝐵𝑀	NOUN
asir-36063	304	3	′	′	NUM
asir-36063	304	4	)	)	PUNCT
asir-36063	304	5	<	<	X
asir-36063	304	6	𝑛	𝑛	PROPN
asir-36063	304	7	2	2	NUM
asir-36063	304	8	(	(	PUNCT
asir-36063	304	9	𝑝	𝑝	PROPN
asir-36063	304	10	−	−	PROPN
asir-36063	304	11	𝑞)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	𝑞)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	304	12	.	.	PUNCT
asir-36063	304	13	13	13	NUM
asir-36063	304	14	)	)	PUNCT
asir-36063	304	15	can	can	AUX
asir-36063	304	16	guarantee	guarantee	VERB
asir-36063	304	17	the	the	DET
asir-36063	304	18	exact	exact	ADJ
asir-36063	304	19	recovery	recovery	NOUN
asir-36063	304	20	of	of	ADP
asir-36063	304	21	dsbm	dsbm	ADJ
asir-36063	304	22	.	.	PUNCT
asir-36063	305	1	proof	proof	NOUN
asir-36063	305	2	for	for	ADP
asir-36063	305	3	lemma	lemma	PROPN
asir-36063	305	4	3.7	3.7	NUM
asir-36063	305	5	lemma	lemma	PROPN
asir-36063	305	6	b.1	b.1	PROPN
asir-36063	305	7	.	.	PUNCT
asir-36063	306	1	recall	recall	PROPN
asir-36063	306	2	definition	definition	NOUN
asir-36063	306	3	𝐴.	𝐴.	PROPN
asir-36063	306	4	3	3	NUM
asir-36063	306	5	,	,	PUNCT
asir-36063	306	6	let	let	VERB
asir-36063	306	7	𝑎	𝑎	NOUN
asir-36063	306	8	,	,	PUNCT
asir-36063	306	9	𝑏	𝑏	PROPN
asir-36063	306	10	and	and	CCONJ
asir-36063	306	11	𝚫′	𝚫′	PROPN
asir-36063	306	12	be	be	VERB
asir-36063	306	13	constants	constant	NOUN
asir-36063	306	14	.	.	PUNCT
asir-36063	307	1	we	we	PRON
asir-36063	307	2	have	have	VERB
asir-36063	307	3	,	,	PUNCT
asir-36063	307	4	⁡⁡⁡⁡𝑇	⁡⁡⁡⁡𝑇	NOUN
asir-36063	307	5	(	(	PUNCT
asir-36063	307	6	𝑛	𝑛	PROPN
asir-36063	307	7	2	2	NUM
asir-36063	307	8	,	,	PUNCT
asir-36063	307	9	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	307	10	)	)	PUNCT
asir-36063	307	11	𝑛	𝑛	NOUN
asir-36063	307	12	,	,	PUNCT
asir-36063	307	13	𝑏log⁡(𝑛	𝑏log⁡(𝑛	NOUN
asir-36063	307	14	)	)	PUNCT
asir-36063	307	15	𝑛	𝑛	PROPN
asir-36063	307	16	,	,	PUNCT
asir-36063	307	17	−δ′√log⁡(𝑛	−δ′√log⁡(𝑛	NOUN
asir-36063	307	18	)	)	PUNCT
asir-36063	307	19	)	)	PUNCT
asir-36063	307	20	≤	≤	NUM
asir-36063	307	21	exp⁡[−	exp⁡[−	NOUN
asir-36063	307	22	(	(	PUNCT
asir-36063	307	23	𝑎	𝑎	X
asir-36063	307	24	+	+	NOUN
asir-36063	307	25	𝑏	𝑏	SYM
asir-36063	307	26	2	2	NUM
asir-36063	307	27	−	−	NOUN
asir-36063	307	28	√𝑎𝑏	√𝑎𝑏	PROPN
asir-36063	307	29	−	−	PROPN
asir-36063	307	30	𝛿(𝑛	𝛿(𝑛	NOUN
asir-36063	307	31	)	)	PUNCT
asir-36063	307	32	)	)	PUNCT
asir-36063	307	33	log⁡(𝑛)]⁡⁡⁡⁡⁡(b	log⁡(𝑛)]⁡⁡⁡⁡⁡(b	PUNCT
asir-36063	307	34	.	.	PUNCT
asir-36063	308	1	14	14	NUM
asir-36063	308	2	)	)	PUNCT
asir-36063	308	3	with	with	ADP
asir-36063	308	4	lim𝑛→∞	lim𝑛→∞	NOUN
asir-36063	308	5	 	 	SPACE
asir-36063	308	6	𝛿(𝑛	𝛿(𝑛	NOUN
asir-36063	308	7	)	)	PUNCT
asir-36063	308	8	proof	proof	NOUN
asir-36063	308	9	of	of	ADP
asir-36063	308	10	this	this	DET
asir-36063	308	11	lemma	lemma	PROPN
asir-36063	308	12	can	can	AUX
asir-36063	308	13	be	be	AUX
asir-36063	308	14	found	find	VERB
asir-36063	308	15	in	in	ADP
asir-36063	308	16	[	[	X
asir-36063	308	17	2	2	NUM
asir-36063	308	18	]	]	PUNCT
asir-36063	308	19	.	.	PUNCT
asir-36063	309	1	we	we	PRON
asir-36063	309	2	are	be	AUX
asir-36063	309	3	now	now	ADV
asir-36063	309	4	ready	ready	ADJ
asir-36063	309	5	to	to	PART
asir-36063	309	6	prove	prove	VERB
asir-36063	309	7	lemma	lemma	PROPN
asir-36063	309	8	3.7	3.7	NUM
asir-36063	309	9	let	let	VERB
asir-36063	309	10	𝑎	𝑎	X
asir-36063	309	11	>	>	X
asir-36063	309	12	𝑏	𝑏	PRON
asir-36063	309	13	be	be	AUX
asir-36063	309	14	constants	constant	NOUN
asir-36063	309	15	and	and	CCONJ
asir-36063	309	16	satisfy	satisfy	VERB
asir-36063	309	17	√𝑎	√𝑎	ADV
asir-36063	309	18	−	−	PROPN
asir-36063	309	19	√𝑏	√𝑏	X
asir-36063	309	20	>	>	X
asir-36063	310	1	√2	√2	PROPN
asir-36063	310	2	.	.	PROPN
asir-36063	310	3	given	give	VERB
asir-36063	310	4	δ	δ	PROPN
asir-36063	310	5	>	>	X
asir-36063	310	6	0	0	PROPN
asir-36063	310	7	,	,	PUNCT
asir-36063	310	8	we	we	PRON
asir-36063	310	9	want	want	VERB
asir-36063	310	10	to	to	PART
asir-36063	310	11	prove	prove	VERB
asir-36063	310	12	that	that	SCONJ
asir-36063	310	13	with	with	ADP
asir-36063	310	14	high	high	ADJ
asir-36063	310	15	probability	probability	NOUN
asir-36063	310	16	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	PROPN
asir-36063	310	17	min	min	NOUN
asir-36063	310	18	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	310	19	]	]	PUNCT
asir-36063	310	20	 	 	SPACE
asir-36063	310	21	(	(	PUNCT
asir-36063	310	22	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	310	23	+	+	CCONJ
asir-36063	310	24	−	−	PROPN
asir-36063	310	25	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	310	26	−	−	NOUN
asir-36063	310	27	)	)	PUNCT
asir-36063	310	28	≥	≥	NOUN
asir-36063	310	29	𝚫	𝚫	PROPN
asir-36063	310	30	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	310	31	)	)	PUNCT
asir-36063	310	32	𝔼[deg𝐶	𝔼[deg𝐶	NOUN
asir-36063	310	33	+	+	NOUN
asir-36063	310	34	⁡(𝑖	⁡(𝑖	NOUN
asir-36063	310	35	)	)	PUNCT
asir-36063	311	1	−	−	ADP
asir-36063	311	2	deg𝐶	deg𝐶	ADJ
asir-36063	311	3	−⁡(𝑖	−⁡(𝑖	PROPN
asir-36063	311	4	)	)	PUNCT
asir-36063	311	5	]	]	PUNCT
asir-36063	312	1	=	=	PUNCT
asir-36063	312	2	𝚫	𝚫	PROPN
asir-36063	312	3	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	312	4	)	)	PUNCT
asir-36063	312	5	𝑛	𝑛	PRON
asir-36063	312	6	2	2	NUM
asir-36063	312	7	(	(	PUNCT
asir-36063	312	8	𝑝	𝑝	PROPN
asir-36063	312	9	−	−	PROPN
asir-36063	312	10	𝑞)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	𝑞)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	312	11	.	.	PROPN
asir-36063	312	12	15	15	NUM
asir-36063	312	13	)	)	PUNCT
asir-36063	312	14	for	for	ADP
asir-36063	312	15	fixed	fix	VERB
asir-36063	312	16	𝑖	𝑖	SYM
asir-36063	312	17	throughout	throughout	ADP
asir-36063	312	18	the	the	DET
asir-36063	312	19	proof	proof	NOUN
asir-36063	312	20	.	.	PUNCT
asir-36063	313	1	we	we	PRON
asir-36063	313	2	can	can	AUX
asir-36063	313	3	write	write	VERB
asir-36063	313	4	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	313	5	applied	apply	VERB
asir-36063	313	6	science	science	NOUN
asir-36063	313	7	and	and	CCONJ
asir-36063	313	8	innovative	innovative	ADJ
asir-36063	313	9	research	research	NOUN
asir-36063	313	10	vol	vol	NOUN
asir-36063	313	11	.	.	PROPN
asir-36063	314	1	8	8	NUM
asir-36063	314	2	,	,	PUNCT
asir-36063	314	3	no	no	INTJ
asir-36063	314	4	.	.	NOUN
asir-36063	314	5	1	1	NUM
asir-36063	314	6	,	,	PUNCT
asir-36063	314	7	2024	2024	NUM
asir-36063	314	8	76	76	NUM
asir-36063	314	9	published	publish	VERB
asir-36063	314	10	by	by	ADP
asir-36063	314	11	scholink	scholink	PROPN
asir-36063	314	12	inc	inc	PROPN
asir-36063	314	13	.	.	PROPN
asir-36063	315	1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(𝐷+)𝑖𝑖	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(𝐷+)𝑖𝑖	PROPN
asir-36063	316	1	−	−	PROPN
asir-36063	317	1	(	(	PUNCT
asir-36063	317	2	𝐷−)𝑖𝑖	𝐷−)𝑖𝑖	NOUN
asir-36063	317	3	=	=	SYM
asir-36063	317	4	(	(	PUNCT
asir-36063	317	5	∑	∑	ADV
asir-36063	317	6	  	  	SPACE
asir-36063	317	7	𝑛/2−1	𝑛/2−1	NUM
asir-36063	317	8	𝑖=1	𝑖=1	PROPN
asir-36063	317	9	 	 	SPACE
asir-36063	317	10	𝑊𝑖	𝑊𝑖	PROPN
asir-36063	317	11	)	)	PUNCT
asir-36063	317	12	−	−	PROPN
asir-36063	317	13	(	(	PUNCT
asir-36063	317	14	∑	∑	PROPN
asir-36063	317	15	  	  	SPACE
asir-36063	317	16	𝑛/2	𝑛/2	X
asir-36063	317	17	𝑖=1	𝑖=1	PUNCT
asir-36063	317	18	 	 	SPACE
asir-36063	317	19	𝑍𝑖	𝑍𝑖	PROPN
asir-36063	317	20	)	)	PUNCT
asir-36063	317	21	=	=	PUNCT
asir-36063	317	22	∑	∑	PUNCT
asir-36063	317	23	  	  	SPACE
asir-36063	317	24	𝑛/2−1	𝑛/2−1	PRON
asir-36063	317	25	𝑖=1	𝑖=1	PROPN
asir-36063	317	26	(	(	PUNCT
asir-36063	317	27	𝑊𝑖	𝑊𝑖	PROPN
asir-36063	317	28	−	−	PROPN
asir-36063	317	29	𝑍𝑖	𝑍𝑖	PROPN
asir-36063	317	30	)	)	PUNCT
asir-36063	317	31	+	+	PROPN
asir-36063	318	1	𝑍𝑛/2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	𝑍𝑛/2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	PROPN
asir-36063	318	2	.	.	PROPN
asir-36063	318	3	16	16	NUM
asir-36063	318	4	)	)	PUNCT
asir-36063	318	5	hence	hence	ADV
asir-36063	318	6	,	,	PUNCT
asir-36063	318	7	we	we	PRON
asir-36063	318	8	substitute	substitute	VERB
asir-36063	318	9	𝑝	𝑝	PROPN
asir-36063	318	10	and	and	CCONJ
asir-36063	318	11	𝑞	𝑞	X
asir-36063	318	12	with	with	ADP
asir-36063	318	13	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	318	14	)	)	PUNCT
asir-36063	318	15	𝑛	𝑛	NOUN
asir-36063	318	16	and	and	CCONJ
asir-36063	318	17	𝑏log⁡(𝑛	𝑏log⁡(𝑛	NOUN
asir-36063	318	18	)	)	PUNCT
asir-36063	318	19	𝑛	𝑛	VERB
asir-36063	318	20	respectively	respectively	ADV
asir-36063	318	21	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	PROPN
asir-36063	318	22	𝚫	𝚫	PROPN
asir-36063	318	23	√log⁡(𝑛	√log⁡(𝑛	PROPN
asir-36063	318	24	)	)	PUNCT
asir-36063	318	25	(	(	PUNCT
asir-36063	318	26	𝑛	𝑛	PROPN
asir-36063	318	27	2	2	NUM
asir-36063	318	28	(	(	PUNCT
asir-36063	318	29	𝑝	𝑝	PROPN
asir-36063	318	30	−	−	PROPN
asir-36063	318	31	𝑞	𝑞	NOUN
asir-36063	318	32	)	)	PUNCT
asir-36063	318	33	)	)	PUNCT
asir-36063	318	34	=	=	SYM
asir-36063	318	35	δ√log⁡(𝑛	δ√log⁡(𝑛	NOUN
asir-36063	318	36	)	)	PUNCT
asir-36063	318	37	(	(	PUNCT
asir-36063	318	38	𝑎	𝑎	X
asir-36063	318	39	−	−	NOUN
asir-36063	318	40	𝑏	𝑏	SYM
asir-36063	318	41	2	2	NUM
asir-36063	318	42	)	)	PUNCT
asir-36063	318	43	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	318	44	.	.	PROPN
asir-36063	319	1	17	17	NUM
asir-36063	319	2	)	)	PUNCT
asir-36063	319	3	we	we	PRON
asir-36063	319	4	have	have	VERB
asir-36063	319	5	the	the	DET
asir-36063	319	6	probability	probability	NOUN
asir-36063	319	7	of	of	ADP
asir-36063	319	8	degin	degin	PROPN
asir-36063	319	9	⁡(𝑖	⁡(𝑖	PROPN
asir-36063	319	10	)	)	PUNCT
asir-36063	320	1	−	−	NOUN
asir-36063	320	2	degout	degout	NOUN
asir-36063	320	3	⁡(𝑖	⁡(𝑖	PUNCT
asir-36063	320	4	)	)	PUNCT
asir-36063	321	1	<	<	X
asir-36063	321	2	𝚫	𝚫	PROPN
asir-36063	321	3	√log⁡(𝑛	√log⁡(𝑛	PROPN
asir-36063	321	4	)	)	PUNCT
asir-36063	321	5	(	(	PUNCT
asir-36063	321	6	𝑛/2(𝑝	𝑛/2(𝑝	VERB
asir-36063	321	7	−	−	PROPN
asir-36063	321	8	𝑞	𝑞	NOUN
asir-36063	321	9	)	)	PUNCT
asir-36063	321	10	)	)	PUNCT
asir-36063	321	11	is	be	AUX
asir-36063	321	12	equal	equal	ADJ
asir-36063	321	13	to	to	ADP
asir-36063	321	14	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	ADJ
asir-36063	321	15	ℙ	ℙ	PROPN
asir-36063	321	16	(	(	PUNCT
asir-36063	321	17	∑	∑	DET
asir-36063	321	18	  	  	SPACE
asir-36063	321	19	𝑛/2−1	𝑛/2−1	NUM
asir-36063	321	20	𝑖=1	𝑖=1	PROPN
asir-36063	321	21	  	  	SPACE
asir-36063	321	22	(	(	PUNCT
asir-36063	321	23	𝑊𝑖	𝑊𝑖	PROPN
asir-36063	321	24	−	−	PROPN
asir-36063	321	25	𝑍𝑖	𝑍𝑖	PROPN
asir-36063	321	26	)	)	PUNCT
asir-36063	321	27	+	+	CCONJ
asir-36063	321	28	𝑍𝑛/2	𝑍𝑛/2	X
asir-36063	321	29	<	<	X
asir-36063	321	30	δ√log⁡(𝑛	δ√log⁡(𝑛	NOUN
asir-36063	321	31	)	)	PUNCT
asir-36063	321	32	(	(	PUNCT
asir-36063	321	33	𝑎	𝑎	X
asir-36063	321	34	−	−	NOUN
asir-36063	321	35	𝑏	𝑏	NOUN
asir-36063	321	36	2	2	NUM
asir-36063	321	37	)	)	PUNCT
asir-36063	321	38	)	)	PUNCT
asir-36063	322	1	=	=	SYM
asir-36063	322	2	ℙ	ℙ	PROPN
asir-36063	322	3	(	(	PUNCT
asir-36063	322	4	∑	∑	DET
asir-36063	322	5	  	  	SPACE
asir-36063	322	6	𝑛/2−1	𝑛/2−1	NUM
asir-36063	322	7	𝑖=1	𝑖=1	PROPN
asir-36063	322	8	  	  	SPACE
asir-36063	322	9	(	(	PUNCT
asir-36063	322	10	𝑍𝑖	𝑍𝑖	PROPN
asir-36063	322	11	−𝑊𝑖	−𝑊𝑖	NOUN
asir-36063	322	12	)	)	PUNCT
asir-36063	322	13	−	−	PROPN
asir-36063	322	14	𝑍𝑛/2	𝑍𝑛/2	PRON
asir-36063	322	15	>	>	X
asir-36063	322	16	−δ√log⁡(𝑛	−δ√log⁡(𝑛	NOUN
asir-36063	322	17	)	)	PUNCT
asir-36063	322	18	(	(	PUNCT
asir-36063	322	19	𝑎	𝑎	X
asir-36063	322	20	−	−	NOUN
asir-36063	322	21	𝑏	𝑏	NOUN
asir-36063	322	22	2	2	NUM
asir-36063	322	23	)	)	PUNCT
asir-36063	322	24	)	)	PUNCT
asir-36063	323	1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	323	2	.	.	PUNCT
asir-36063	324	1	18	18	NUM
asir-36063	324	2	)	)	PUNCT
asir-36063	324	3	which	which	PRON
asir-36063	324	4	is	be	AUX
asir-36063	324	5	upper	upper	ADJ
asir-36063	324	6	bounded	bound	VERB
asir-36063	324	7	by	by	ADP
asir-36063	324	8	,	,	PUNCT
asir-36063	324	9	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ℙ	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ℙ	PROPN
asir-36063	325	1	[	[	X
asir-36063	325	2	∑	∑	INTJ
asir-36063	325	3	  	  	SPACE
asir-36063	325	4	𝑛/2	𝑛/2	X
asir-36063	325	5	𝑖=1	𝑖=1	PUNCT
asir-36063	325	6	  	  	SPACE
asir-36063	325	7	(	(	PUNCT
asir-36063	325	8	𝑍𝑖	𝑍𝑖	PROPN
asir-36063	325	9	−𝑊𝑖	−𝑊𝑖	NOUN
asir-36063	325	10	)	)	PUNCT
asir-36063	325	11	>	>	X
asir-36063	325	12	−δ√log⁡(𝑛	−δ√log⁡(𝑛	NOUN
asir-36063	325	13	)	)	PUNCT
asir-36063	325	14	(	(	PUNCT
asir-36063	325	15	𝑎	𝑎	X
asir-36063	325	16	−	−	NOUN
asir-36063	325	17	𝑏	𝑏	NOUN
asir-36063	325	18	2	2	NUM
asir-36063	325	19	)	)	PUNCT
asir-36063	325	20	]	]	X
asir-36063	325	21	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	325	22	.	.	PUNCT
asir-36063	326	1	19	19	NUM
asir-36063	326	2	)	)	PUNCT
asir-36063	326	3	we	we	PRON
asir-36063	326	4	let	let	VERB
asir-36063	326	5	𝚫′	𝚫′	PROPN
asir-36063	326	6	=	=	SYM
asir-36063	326	7	𝚫	𝚫	PROPN
asir-36063	326	8	(	(	PUNCT
asir-36063	326	9	𝑎−𝑏	𝑎−𝑏	NOUN
asir-36063	326	10	2	2	NUM
asir-36063	326	11	)	)	PUNCT
asir-36063	327	1	+	+	CCONJ
asir-36063	327	2	1	1	NUM
asir-36063	327	3	,	,	PUNCT
asir-36063	327	4	then	then	ADV
asir-36063	327	5	using	use	VERB
asir-36063	327	6	the	the	DET
asir-36063	327	7	previous	previous	ADJ
asir-36063	327	8	definition	definition	NOUN
asir-36063	327	9	,	,	PUNCT
asir-36063	327	10	we	we	PRON
asir-36063	327	11	can	can	AUX
asir-36063	327	12	obtain	obtain	VERB
asir-36063	327	13	the	the	DET
asir-36063	327	14	following	follow	VERB
asir-36063	327	15	inequalities	inequality	NOUN
asir-36063	327	16	:	:	PUNCT
asir-36063	327	17	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ℙ((𝒟𝑖𝑖	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ℙ((𝒟𝑖𝑖	X
asir-36063	327	18	+	+	NUM
asir-36063	327	19	−𝒟𝑖𝑖	−𝒟𝑖𝑖	ADP
asir-36063	327	20	−	−	NOUN
asir-36063	327	21	)	)	PUNCT
asir-36063	327	22	<	<	X
asir-36063	327	23	𝚫	𝚫	PROPN
asir-36063	327	24	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	327	25	)	)	PUNCT
asir-36063	327	26	𝔼[deg𝐶	𝔼[deg𝐶	NOUN
asir-36063	327	27	+	+	NOUN
asir-36063	327	28	⁡(𝑖	⁡(𝑖	NOUN
asir-36063	327	29	)	)	PUNCT
asir-36063	328	1	−	−	ADP
asir-36063	328	2	deg𝐶	deg𝐶	ADJ
asir-36063	328	3	−⁡(𝑖	−⁡(𝑖	PROPN
asir-36063	328	4	)	)	PUNCT
asir-36063	328	5	]	]	PUNCT
asir-36063	328	6	)	)	PUNCT
asir-36063	328	7	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡≤	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡≤	CCONJ
asir-36063	328	8	𝑇	𝑇	PROPN
asir-36063	328	9	(	(	PUNCT
asir-36063	328	10	𝑛	𝑛	DET
asir-36063	328	11	2	2	NUM
asir-36063	328	12	,	,	PUNCT
asir-36063	328	13	𝑎log⁡(𝑛	𝑎log⁡(𝑛	NOUN
asir-36063	328	14	)	)	PUNCT
asir-36063	328	15	𝑛	𝑛	NOUN
asir-36063	328	16	,	,	PUNCT
asir-36063	328	17	𝑏log⁡(𝑛	𝑏log⁡(𝑛	NOUN
asir-36063	328	18	)	)	PUNCT
asir-36063	328	19	𝑛	𝑛	PROPN
asir-36063	328	20	,	,	PUNCT
asir-36063	328	21	−𝚫′√log⁡(𝑛	−𝚫′√log⁡(𝑛	PROPN
asir-36063	328	22	)	)	PUNCT
asir-36063	328	23	)	)	PUNCT
asir-36063	329	1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡≤	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡≤	ADV
asir-36063	329	2	exp⁡[−	exp⁡[−	NOUN
asir-36063	329	3	(	(	PUNCT
asir-36063	329	4	𝑎	𝑎	X
asir-36063	329	5	+	+	NOUN
asir-36063	329	6	𝑏	𝑏	SYM
asir-36063	329	7	2	2	NUM
asir-36063	329	8	−	−	NOUN
asir-36063	329	9	√𝑎𝑏	√𝑎𝑏	PROPN
asir-36063	329	10	−	−	PROPN
asir-36063	329	11	𝛿(𝑛	𝛿(𝑛	NOUN
asir-36063	329	12	)	)	PUNCT
asir-36063	329	13	)	)	PUNCT
asir-36063	329	14	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	329	15	)	)	PUNCT
asir-36063	329	16	]	]	PUNCT
asir-36063	329	17	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	ADJ
asir-36063	329	18	.	.	PUNCT
asir-36063	330	1	20	20	NUM
asir-36063	330	2	)	)	PUNCT
asir-36063	330	3	by	by	ADP
asir-36063	330	4	using	use	VERB
asir-36063	330	5	union	union	NOUN
asir-36063	330	6	bound	bind	VERB
asir-36063	330	7	,	,	PUNCT
asir-36063	330	8	we	we	PRON
asir-36063	330	9	can	can	AUX
asir-36063	330	10	have	have	VERB
asir-36063	330	11	:	:	PUNCT
asir-36063	330	12	⁡⁡⁡ℙ	⁡⁡⁡ℙ	ADP
asir-36063	330	13	[	[	PUNCT
asir-36063	330	14	min	min	NOUN
asir-36063	330	15	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	330	16	]	]	PUNCT
asir-36063	330	17	 	 	SPACE
asir-36063	330	18	(	(	PUNCT
asir-36063	330	19	𝑫𝑖𝑖	𝑫𝑖𝑖	PROPN
asir-36063	330	20	+	+	PROPN
asir-36063	330	21	−𝑫𝑖𝑖	−𝑫𝑖𝑖	NOUN
asir-36063	330	22	−	−	NOUN
asir-36063	330	23	<	<	X
asir-36063	330	24	𝚫	𝚫	PROPN
asir-36063	330	25	√log⁡(𝑛	√log⁡(𝑛	PROPN
asir-36063	330	26	)	)	PUNCT
asir-36063	330	27	𝑛	𝑛	PRON
asir-36063	330	28	2	2	NUM
asir-36063	330	29	(	(	PUNCT
asir-36063	330	30	𝑝	𝑝	PROPN
asir-36063	330	31	−	−	PROPN
asir-36063	330	32	𝑞	𝑞	PROPN
asir-36063	330	33	)	)	PUNCT
asir-36063	330	34	)	)	PUNCT
asir-36063	330	35	]	]	PUNCT
asir-36063	331	1	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡≤	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡≤	NUM
asir-36063	331	2	exp⁡[−	exp⁡[−	NOUN
asir-36063	331	3	(	(	PUNCT
asir-36063	331	4	𝑎	𝑎	X
asir-36063	331	5	+	+	NOUN
asir-36063	331	6	𝑏	𝑏	SYM
asir-36063	331	7	2	2	NUM
asir-36063	331	8	−	−	NOUN
asir-36063	331	9	√𝑎𝑏	√𝑎𝑏	NOUN
asir-36063	331	10	−	−	PROPN
asir-36063	331	11	1	1	NUM
asir-36063	331	12	−	−	PROPN
asir-36063	331	13	𝛿(𝑛	𝛿(𝑛	NOUN
asir-36063	331	14	)	)	PUNCT
asir-36063	331	15	)	)	PUNCT
asir-36063	331	16	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	331	17	)	)	PUNCT
asir-36063	331	18	]	]	PUNCT
asir-36063	331	19	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b21	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b21	NOUN
asir-36063	331	20	)	)	PUNCT
asir-36063	331	21	from	from	ADP
asir-36063	331	22	this	this	PRON
asir-36063	331	23	,	,	PUNCT
asir-36063	331	24	if	if	SCONJ
asir-36063	331	25	we	we	PRON
asir-36063	331	26	have	have	VERB
asir-36063	331	27	𝑎+𝑏	𝑎+𝑏	NUM
asir-36063	331	28	2	2	NUM
asir-36063	331	29	−	−	NOUN
asir-36063	331	30	√𝑎𝑏	√𝑎𝑏	PROPN
asir-36063	331	31	>	>	X
asir-36063	331	32	1	1	NUM
asir-36063	331	33	,	,	PUNCT
asir-36063	331	34	which	which	PRON
asir-36063	331	35	can	can	AUX
asir-36063	331	36	be	be	AUX
asir-36063	331	37	re	re	VERB
asir-36063	331	38	-	-	VERB
asir-36063	331	39	written	write	VERB
asir-36063	331	40	into	into	ADP
asir-36063	331	41	√𝑎	√𝑎	ADJ
asir-36063	331	42	−	−	PROPN
asir-36063	331	43	√𝑏	√𝑏	NOUN
asir-36063	331	44	>	>	X
asir-36063	332	1	√2	√2	NOUN
asir-36063	332	2	,	,	PUNCT
asir-36063	332	3	then	then	ADV
asir-36063	332	4	this	this	PRON
asir-36063	332	5	means	mean	VERB
asir-36063	332	6	that	that	SCONJ
asir-36063	332	7	the	the	DET
asir-36063	332	8	probability	probability	NOUN
asir-36063	332	9	of	of	ADP
asir-36063	332	10	ℙ	ℙ	PROPN
asir-36063	332	11	[	[	X
asir-36063	332	12	min𝑖∈[2𝑛	min𝑖∈[2𝑛	NOUN
asir-36063	332	13	]	]	PUNCT
asir-36063	332	14	 	 	SPACE
asir-36063	332	15	(	(	PUNCT
asir-36063	332	16	𝑫𝑖𝑖	𝑫𝑖𝑖	PROPN
asir-36063	332	17	+	+	PROPN
asir-36063	332	18	−𝑫𝑖𝑖	−𝑫𝑖𝑖	NOUN
asir-36063	332	19	−	−	NOUN
asir-36063	332	20	<	<	X
asir-36063	332	21	𝚫	𝚫	PROPN
asir-36063	332	22	√log⁡(𝑛	√log⁡(𝑛	PROPN
asir-36063	332	23	)	)	PUNCT
asir-36063	332	24	𝑛	𝑛	PRON
asir-36063	332	25	2	2	NUM
asir-36063	332	26	(	(	PUNCT
asir-36063	332	27	𝑝	𝑝	PROPN
asir-36063	332	28	−	−	PROPN
asir-36063	332	29	𝑞	𝑞	PROPN
asir-36063	332	30	)	)	PUNCT
asir-36063	332	31	)	)	PUNCT
asir-36063	332	32	]	]	PUNCT
asir-36063	332	33	is	be	AUX
asir-36063	332	34	negative	negative	ADJ
asir-36063	332	35	.	.	PUNCT
asir-36063	333	1	then	then	ADV
asir-36063	333	2	when	when	SCONJ
asir-36063	333	3	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡√𝑎	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡√𝑎	PUNCT
asir-36063	333	4	−	−	NOUN
asir-36063	333	5	√𝑏	√𝑏	NOUN
asir-36063	333	6	>	>	X
asir-36063	333	7	√2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	√2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	PROPN
asir-36063	333	8	.	.	PROPN
asir-36063	333	9	22	22	NUM
asir-36063	333	10	)	)	PUNCT
asir-36063	333	11	it	it	PRON
asir-36063	333	12	holds	hold	VERB
asir-36063	333	13	true	true	ADJ
asir-36063	333	14	with	with	ADP
asir-36063	333	15	high	high	ADJ
asir-36063	333	16	probability	probability	NOUN
asir-36063	333	17	that	that	SCONJ
asir-36063	333	18	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡	DET
asir-36063	333	19	min	min	NOUN
asir-36063	333	20	𝑖∈[2𝑛	𝑖∈[2𝑛	PROPN
asir-36063	333	21	]	]	PUNCT
asir-36063	333	22	 	 	SPACE
asir-36063	333	23	(	(	PUNCT
asir-36063	333	24	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	333	25	+	+	CCONJ
asir-36063	333	26	−	−	PROPN
asir-36063	333	27	𝒟𝑖𝑖	𝒟𝑖𝑖	PROPN
asir-36063	333	28	−	−	NOUN
asir-36063	333	29	)	)	PUNCT
asir-36063	333	30	≥	≥	NOUN
asir-36063	333	31	𝚫	𝚫	PROPN
asir-36063	333	32	log⁡(𝑛	log⁡(𝑛	PROPN
asir-36063	333	33	)	)	PUNCT
asir-36063	333	34	𝔼[deg𝐶	𝔼[deg𝐶	NOUN
asir-36063	333	35	+	+	NOUN
asir-36063	333	36	⁡(𝑖	⁡(𝑖	NOUN
asir-36063	333	37	)	)	PUNCT
asir-36063	334	1	−	−	ADP
asir-36063	334	2	deg𝐶	deg𝐶	ADJ
asir-36063	334	3	−⁡(𝑖)]⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	−⁡(𝑖)]⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(b	NOUN
asir-36063	334	4	.	.	PUNCT
asir-36063	335	1	23	23	NUM
asir-36063	335	2	)	)	PUNCT
asir-36063	335	3	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-36063	335	4	applied	apply	VERB
asir-36063	335	5	science	science	NOUN
asir-36063	335	6	and	and	CCONJ
asir-36063	335	7	innovative	innovative	ADJ
asir-36063	335	8	research	research	NOUN
asir-36063	335	9	vol	vol	NOUN
asir-36063	335	10	.	.	PROPN
asir-36063	336	1	8	8	NUM
asir-36063	336	2	,	,	PUNCT
asir-36063	336	3	no	no	INTJ
asir-36063	336	4	.	.	NOUN
asir-36063	336	5	1	1	NUM
asir-36063	336	6	,	,	PUNCT
asir-36063	336	7	2024	2024	NUM
asir-36063	336	8	77	77	NUM
asir-36063	336	9	published	publish	VERB
asir-36063	336	10	by	by	ADP
asir-36063	336	11	scholink	scholink	PROPN
asir-36063	336	12	inc	inc	PROPN
asir-36063	336	13	.	.	PROPN
asir-36063	337	1	references	references	PROPN
asir-36063	337	2	afonso	afonso	PROPN
asir-36063	337	3	s.	s.	PROPN
asir-36063	337	4	bandeira	bandeira	PROPN
asir-36063	337	5	.	.	PUNCT
asir-36063	338	1	(	(	PUNCT
asir-36063	338	2	2015	2015	NUM
asir-36063	338	3	)	)	PUNCT
asir-36063	338	4	.	.	PUNCT
asir-36063	339	1	random	random	ADJ
asir-36063	339	2	laplacian	laplacian	ADJ
asir-36063	339	3	matrices	matrix	NOUN
asir-36063	339	4	and	and	CCONJ
asir-36063	339	5	convex	convex	ADJ
asir-36063	339	6	relaxations	relaxation	NOUN
asir-36063	339	7	.	.	PUNCT
asir-36063	340	1	https://doi.org/10.1007/s10208-016-9341-9	https://doi.org/10.1007/s10208-016-9341-9	NOUN
asir-36063	340	2	andrea	andrea	PROPN
asir-36063	340	3	lancichinetti	lancichinetti	PROPN
asir-36063	340	4	,	,	PUNCT
asir-36063	340	5	santo	santo	PROPN
asir-36063	340	6	fortunato	fortunato	PROPN
asir-36063	340	7	,	,	PUNCT
asir-36063	340	8	&	&	CCONJ
asir-36063	340	9	filippo	filippo	PROPN
asir-36063	340	10	radicchi	radicchi	PROPN
asir-36063	340	11	.	.	PROPN
asir-36063	340	12	(	(	PUNCT
asir-36063	340	13	2008	2008	NUM
asir-36063	340	14	)	)	PUNCT
asir-36063	340	15	.	.	PUNCT
asir-36063	341	1	benchmark	benchmark	NOUN
asir-36063	341	2	graphs	graph	NOUN
asir-36063	341	3	for	for	ADP
asir-36063	341	4	testing	test	VERB
asir-36063	341	5	community	community	NOUN
asir-36063	341	6	detection	detection	NOUN
asir-36063	341	7	algorithms	algorithm	NOUN
asir-36063	341	8	.	.	PUNCT
asir-36063	342	1	in	in	ADP
asir-36063	342	2	physical	physical	ADJ
asir-36063	342	3	review	review	NOUN
asir-36063	342	4	e	e	PROPN
asir-36063	342	5	78.4	78.4	NUM
asir-36063	342	6	.	.	PUNCT
asir-36063	343	1	https://doi.org/10.1103/physreve.78.046110	https://doi.org/10.1103/physreve.78.046110	PROPN
asir-36063	343	2	emmanuel	emmanuel	PROPN
asir-36063	343	3	abbe	abbe	PROPN
asir-36063	343	4	,	,	PUNCT
asir-36063	343	5	afonso	afonso	PROPN
asir-36063	343	6	s.	s.	PROPN
asir-36063	343	7	bandeira	bandeira	PROPN
asir-36063	343	8	,	,	PUNCT
asir-36063	343	9	&	&	CCONJ
asir-36063	343	10	georgina	georgina	PROPN
asir-36063	343	11	hall	hall	PROPN
asir-36063	343	12	.	.	PUNCT
asir-36063	344	1	(	(	PUNCT
asir-36063	344	2	2014	2014	NUM
asir-36063	344	3	)	)	PUNCT
asir-36063	344	4	.	.	PUNCT
asir-36063	345	1	exact	exact	ADJ
asir-36063	345	2	recovery	recovery	NOUN
asir-36063	345	3	in	in	ADP
asir-36063	345	4	the	the	DET
asir-36063	345	5	stochastic	stochastic	ADJ
asir-36063	345	6	block	block	NOUN
asir-36063	345	7	model	model	NOUN
asir-36063	345	8	.	.	PUNCT
asir-36063	346	1	in	in	ADP
asir-36063	346	2	corr	corr	PROPN
asir-36063	346	3	abs/1405.3267	abs/1405.3267	PROPN
asir-36063	346	4	.	.	PUNCT
asir-36063	347	1	emmanuel	emmanuel	PROPN
asir-36063	347	2	abbe	abbe	PROPN
asir-36063	347	3	,	,	PUNCT
asir-36063	347	4	afonso	afonso	PROPN
asir-36063	347	5	s.	s.	PROPN
asir-36063	347	6	bandeira	bandeira	PROPN
asir-36063	347	7	,	,	PUNCT
asir-36063	347	8	&	&	CCONJ
asir-36063	347	9	georgina	georgina	PROPN
asir-36063	347	10	hall	hall	PROPN
asir-36063	347	11	.	.	PUNCT
asir-36063	348	1	(	(	PUNCT
asir-36063	348	2	2014	2014	NUM
asir-36063	348	3	)	)	PUNCT
asir-36063	348	4	.	.	PUNCT
asir-36063	349	1	exact	exact	ADJ
asir-36063	349	2	recovery	recovery	NOUN
asir-36063	349	3	in	in	ADP
asir-36063	349	4	the	the	DET
asir-36063	349	5	stochastic	stochastic	ADJ
asir-36063	349	6	block	block	NOUN
asir-36063	349	7	model	model	NOUN
asir-36063	349	8	.	.	PUNCT
asir-36063	350	1	girvan	girvan	PROPN
asir-36063	350	2	,	,	PUNCT
asir-36063	350	3	m.	m.	NOUN
asir-36063	350	4	,	,	PUNCT
asir-36063	350	5	&	&	CCONJ
asir-36063	350	6	newman	newman	PROPN
asir-36063	350	7	,	,	PUNCT
asir-36063	350	8	m.	m.	PROPN
asir-36063	350	9	e.	e.	PROPN
asir-36063	350	10	j.	j.	PROPN
asir-36063	350	11	(	(	PUNCT
asir-36063	350	12	2002	2002	NUM
asir-36063	350	13	)	)	PUNCT
asir-36063	350	14	.	.	PUNCT
asir-36063	351	1	community	community	NOUN
asir-36063	351	2	structure	structure	NOUN
asir-36063	351	3	in	in	ADP
asir-36063	351	4	social	social	ADJ
asir-36063	351	5	and	and	CCONJ
asir-36063	351	6	biological	biological	ADJ
asir-36063	351	7	networks	network	NOUN
asir-36063	351	8	.	.	PUNCT
asir-36063	352	1	in	in	ADP
asir-36063	352	2	proceedings	proceeding	NOUN
asir-36063	352	3	of	of	ADP
asir-36063	352	4	the	the	DET
asir-36063	352	5	national	national	PROPN
asir-36063	352	6	academy	academy	PROPN
asir-36063	352	7	of	of	ADP
asir-36063	352	8	sciences	sciences	PROPN
asir-36063	352	9	99.12	99.12	NUM
asir-36063	352	10	(	(	PUNCT
asir-36063	352	11	pp	pp	ADJ
asir-36063	352	12	.	.	PUNCT
asir-36063	352	13	7821	7821	NUM
asir-36063	352	14	-	-	SYM
asir-36063	352	15	7826	7826	NUM
asir-36063	352	16	)	)	PUNCT
asir-36063	352	17	.	.	PUNCT
asir-36063	353	1	https://doi.org/10.1073/pnas.122653799	https://doi.org/10.1073/pnas.122653799	PROPN
asir-36063	353	2	newman	newman	PROPN
asir-36063	353	3	,	,	PUNCT
asir-36063	353	4	m.	m.	PROPN
asir-36063	353	5	e.	e.	PROPN
asir-36063	353	6	j.	j.	PROPN
asir-36063	353	7	(	(	PUNCT
asir-36063	353	8	2003	2003	NUM
asir-36063	353	9	)	)	PUNCT
asir-36063	353	10	.	.	PUNCT
asir-36063	354	1	the	the	DET
asir-36063	354	2	structure	structure	NOUN
asir-36063	354	3	and	and	CCONJ
asir-36063	354	4	function	function	NOUN
asir-36063	354	5	of	of	ADP
asir-36063	354	6	complex	complex	ADJ
asir-36063	354	7	networks	network	NOUN
asir-36063	354	8	.	.	PUNCT
asir-36063	355	1	in	in	ADP
asir-36063	355	2	siam	siam	PROPN
asir-36063	355	3	review	review	NOUN
asir-36063	355	4	45.2	45.2	NUM
asir-36063	355	5	(	(	PUNCT
asir-36063	355	6	pp	pp	ADJ
asir-36063	355	7	.	.	PUNCT
asir-36063	355	8	167	167	NUM
asir-36063	355	9	-	-	SYM
asir-36063	355	10	256	256	NUM
asir-36063	355	11	)	)	PUNCT
asir-36063	355	12	.	.	PUNCT
asir-36063	356	1	https://doi.org/10.1137/s003614450342480	https://doi.org/10.1137/s003614450342480	PROPN
asir-36063	356	2	santo	santo	PROPN
asir-36063	356	3	fortunato	fortunato	PROPN
asir-36063	356	4	.	.	PUNCT
asir-36063	356	5	(	(	PUNCT
asir-36063	356	6	2010	2010	NUM
asir-36063	356	7	)	)	PUNCT
asir-36063	356	8	.	.	PUNCT
asir-36063	357	1	community	community	NOUN
asir-36063	357	2	detection	detection	NOUN
asir-36063	357	3	in	in	ADP
asir-36063	357	4	graphs	graph	NOUN
asir-36063	357	5	.	.	PUNCT
asir-36063	358	1	in	in	ADP
asir-36063	358	2	physics	physics	NOUN
asir-36063	358	3	reports	report	VERB
asir-36063	358	4	486.3	486.3	NUM
asir-36063	358	5	-	-	SYM
asir-36063	358	6	5	5	NUM
asir-36063	358	7	(	(	PUNCT
asir-36063	358	8	pp	pp	ADJ
asir-36063	358	9	.	.	PUNCT
asir-36063	359	1	75	75	NUM
asir-36063	359	2	-	-	SYM
asir-36063	359	3	174	174	NUM
asir-36063	359	4	)	)	PUNCT
asir-36063	359	5	.	.	PUNCT
asir-36063	360	1	https://doi.org/10.1016/j.physrep.2009.11.002	https://doi.org/10.1016/j.physrep.2009.11.002	ADJ