id	sid	tid	token	lemma	pos
cana-5767	1	1	communications	communication	NOUN
cana-5767	1	2	on	on	ADP
cana-5767	1	3	applied	apply	VERB
cana-5767	1	4	nonlinear	nonlinear	ADJ
cana-5767	1	5	analysis	analysis	NOUN
cana-5767	1	6	issn	issn	NOUN
cana-5767	1	7	:	:	PUNCT
cana-5767	1	8	1074	1074	NUM
cana-5767	1	9	-	-	PUNCT
cana-5767	1	10	133x	133x	NUM
cana-5767	1	11	vol	vol	NOUN
cana-5767	1	12	32	32	NUM
cana-5767	1	13	no	no	NOUN
cana-5767	1	14	.	.	NOUN
cana-5767	1	15	1	1	NUM
cana-5767	1	16	(	(	PUNCT
cana-5767	1	17	2025	2025	NUM
cana-5767	1	18	)	)	PUNCT
cana-5767	2	1	559	559	NUM
cana-5767	2	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-5767	2	3	exploring	explore	VERB
cana-5767	2	4	the	the	DET
cana-5767	2	5	essential	essential	ADJ
cana-5767	2	6	spectrum	spectrum	NOUN
cana-5767	2	7	extensions	extension	NOUN
cana-5767	2	8	of	of	ADP
cana-5767	2	9	weyl	weyl	PROPN
cana-5767	2	10	's	's	PART
cana-5767	2	11	theorem	theorem	NOUN
cana-5767	2	12	and	and	CCONJ
cana-5767	2	13	their	their	PRON
cana-5767	2	14	applications	application	NOUN
cana-5767	2	15	in	in	ADP
cana-5767	2	16	machine	machine	NOUN
cana-5767	2	17	learning	learn	VERB
cana-5767	2	18	s.	s.	PROPN
cana-5767	2	19	usha	usha	PROPN
cana-5767	2	20	,	,	PUNCT
cana-5767	2	21	assistant	assistant	NOUN
cana-5767	2	22	professor	professor	NOUN
cana-5767	2	23	,	,	PUNCT
cana-5767	2	24	sri	sri	PROPN
cana-5767	2	25	shakthi	shakthi	PROPN
cana-5767	2	26	institute	institute	PROPN
cana-5767	2	27	of	of	ADP
cana-5767	2	28	engineering	engineering	NOUN
cana-5767	2	29	and	and	CCONJ
cana-5767	2	30	technology	technology	NOUN
cana-5767	2	31	,	,	PUNCT
cana-5767	2	32	1082usha@gmail.com	1082usha@gmail.com	NUM
cana-5767	2	33	.	.	PUNCT
cana-5767	3	1	d.senthilkumar	d.senthilkumar	PROPN
cana-5767	3	2	,	,	PUNCT
cana-5767	3	3	post	post	ADJ
cana-5767	3	4	-	-	ADJ
cana-5767	3	5	graduate	graduate	ADJ
cana-5767	3	6	and	and	CCONJ
cana-5767	3	7	research	research	NOUN
cana-5767	3	8	department	department	PROPN
cana-5767	3	9	of	of	ADP
cana-5767	3	10	mathematics	mathematic	NOUN
cana-5767	3	11	,	,	PUNCT
cana-5767	3	12	government	government	NOUN
cana-5767	3	13	arts	arts	PROPN
cana-5767	3	14	college	college	PROPN
cana-5767	3	15	(	(	PUNCT
cana-5767	3	16	autonomous	autonomous	ADJ
cana-5767	3	17	)	)	PUNCT
cana-5767	3	18	,	,	PUNCT
cana-5767	3	19	coimbatore	coimbatore	PROPN
cana-5767	3	20	.	.	PUNCT
cana-5767	4	1	article	article	NOUN
cana-5767	4	2	history	history	NOUN
cana-5767	4	3	:	:	PUNCT
cana-5767	4	4	received	receive	VERB
cana-5767	4	5	:	:	PUNCT
cana-5767	4	6	08	08	NUM
cana-5767	4	7	-	-	SYM
cana-5767	4	8	11	11	NUM
cana-5767	4	9	-	-	PUNCT
cana-5767	4	10	2024	2024	NUM
cana-5767	4	11	revised	revise	VERB
cana-5767	4	12	:	:	PUNCT
cana-5767	4	13	23	23	NUM
cana-5767	4	14	-	-	SYM
cana-5767	4	15	12	12	NUM
cana-5767	4	16	-	-	PUNCT
cana-5767	4	17	2024	2024	NUM
cana-5767	4	18	accepted	accept	VERB
cana-5767	4	19	:	:	PUNCT
cana-5767	4	20	09	09	NUM
cana-5767	4	21	-	-	SYM
cana-5767	4	22	01	01	NUM
cana-5767	4	23	-	-	PUNCT
cana-5767	4	24	2025	2025	NUM
cana-5767	4	25	abstract	abstract	NOUN
cana-5767	4	26	this	this	DET
cana-5767	4	27	paper	paper	NOUN
cana-5767	4	28	examines	examine	VERB
cana-5767	4	29	the	the	DET
cana-5767	4	30	extensions	extension	NOUN
cana-5767	4	31	of	of	ADP
cana-5767	4	32	weyl	weyl	PROPN
cana-5767	4	33	’s	’s	PART
cana-5767	4	34	theorem	theorem	NOUN
cana-5767	4	35	and	and	CCONJ
cana-5767	4	36	their	their	PRON
cana-5767	4	37	relevance	relevance	NOUN
cana-5767	4	38	in	in	ADP
cana-5767	4	39	machine	machine	NOUN
cana-5767	4	40	learning	learning	NOUN
cana-5767	4	41	.	.	PUNCT
cana-5767	5	1	by	by	ADP
cana-5767	5	2	ensuring	ensure	VERB
cana-5767	5	3	the	the	DET
cana-5767	5	4	stability	stability	NOUN
cana-5767	5	5	of	of	ADP
cana-5767	5	6	the	the	DET
cana-5767	5	7	essential	essential	ADJ
cana-5767	5	8	spectrum	spectrum	NOUN
cana-5767	5	9	under	under	ADP
cana-5767	5	10	perturbations	perturbation	NOUN
cana-5767	5	11	,	,	PUNCT
cana-5767	5	12	weyl	weyl	PROPN
cana-5767	5	13	’s	’s	PART
cana-5767	5	14	theorem	theorem	NOUN
cana-5767	5	15	provides	provide	VERB
cana-5767	5	16	a	a	DET
cana-5767	5	17	theoretical	theoretical	ADJ
cana-5767	5	18	foundation	foundation	NOUN
cana-5767	5	19	for	for	ADP
cana-5767	5	20	robust	robust	ADJ
cana-5767	5	21	algorithms	algorithm	NOUN
cana-5767	5	22	,	,	PUNCT
cana-5767	5	23	including	include	VERB
cana-5767	5	24	spectral	spectral	ADJ
cana-5767	5	25	clustering	clustering	NOUN
cana-5767	5	26	,	,	PUNCT
cana-5767	5	27	kernel	kernel	NOUN
cana-5767	5	28	methods	method	NOUN
cana-5767	5	29	,	,	PUNCT
cana-5767	5	30	and	and	CCONJ
cana-5767	5	31	graph	graph	NOUN
cana-5767	5	32	-	-	PUNCT
cana-5767	5	33	based	base	VERB
cana-5767	5	34	learning	learning	NOUN
cana-5767	5	35	.	.	PUNCT
cana-5767	6	1	we	we	PRON
cana-5767	6	2	present	present	VERB
cana-5767	6	3	key	key	ADJ
cana-5767	6	4	theorems	theorem	NOUN
cana-5767	6	5	,	,	PUNCT
cana-5767	6	6	illustrative	illustrative	ADJ
cana-5767	6	7	examples	example	NOUN
cana-5767	6	8	,	,	PUNCT
cana-5767	6	9	and	and	CCONJ
cana-5767	6	10	computational	computational	ADJ
cana-5767	6	11	experiments	experiment	NOUN
cana-5767	6	12	to	to	PART
cana-5767	6	13	demonstrate	demonstrate	VERB
cana-5767	6	14	the	the	DET
cana-5767	6	15	practical	practical	ADJ
cana-5767	6	16	impact	impact	NOUN
cana-5767	6	17	of	of	ADP
cana-5767	6	18	spectral	spectral	ADJ
cana-5767	6	19	stability	stability	NOUN
cana-5767	6	20	on	on	ADP
cana-5767	6	21	data	data	NOUN
cana-5767	6	22	-	-	PUNCT
cana-5767	6	23	driven	drive	VERB
cana-5767	6	24	models	model	NOUN
cana-5767	6	25	.	.	PUNCT
cana-5767	7	1	keywords	keyword	NOUN
cana-5767	7	2	;	;	PUNCT
cana-5767	7	3	weyl	weyl	PROPN
cana-5767	7	4	’s	’s	PART
cana-5767	7	5	theorem	theorem	PROPN
cana-5767	7	6	,	,	PUNCT
cana-5767	7	7	spectral	spectral	ADJ
cana-5767	7	8	stability	stability	NOUN
cana-5767	7	9	,	,	PUNCT
cana-5767	7	10	essential	essential	ADJ
cana-5767	7	11	spectrum	spectrum	NOUN
cana-5767	7	12	,	,	PUNCT
cana-5767	7	13	operator	operator	NOUN
cana-5767	7	14	theory	theory	NOUN
cana-5767	7	15	machine	machine	NOUN
cana-5767	7	16	learning	learning	PROPN
cana-5767	7	17	,	,	PUNCT
cana-5767	7	18	spectral	spectral	ADJ
cana-5767	7	19	clustering	clustering	ADJ
cana-5767	7	20	introduction	introduction	NOUN
cana-5767	7	21	spectral	spectral	ADJ
cana-5767	7	22	methods	method	NOUN
cana-5767	7	23	have	have	AUX
cana-5767	7	24	become	become	VERB
cana-5767	7	25	indispensable	indispensable	ADJ
cana-5767	7	26	tools	tool	NOUN
cana-5767	7	27	in	in	ADP
cana-5767	7	28	modern	modern	ADJ
cana-5767	7	29	machine	machine	NOUN
cana-5767	7	30	learning	learning	NOUN
cana-5767	7	31	due	due	ADP
cana-5767	7	32	to	to	ADP
cana-5767	7	33	their	their	PRON
cana-5767	7	34	ability	ability	NOUN
cana-5767	7	35	to	to	PART
cana-5767	7	36	reveal	reveal	VERB
cana-5767	7	37	meaningful	meaningful	ADJ
cana-5767	7	38	structures	structure	NOUN
cana-5767	7	39	in	in	ADP
cana-5767	7	40	data	datum	NOUN
cana-5767	7	41	through	through	ADP
cana-5767	7	42	eigenvalues	eigenvalue	NOUN
cana-5767	7	43	and	and	CCONJ
cana-5767	7	44	eigenvectors	eigenvector	NOUN
cana-5767	7	45	of	of	ADP
cana-5767	7	46	matrices	matrix	NOUN
cana-5767	7	47	.	.	PUNCT
cana-5767	8	1	these	these	DET
cana-5767	8	2	methods	method	NOUN
cana-5767	8	3	find	find	VERB
cana-5767	8	4	applications	application	NOUN
cana-5767	8	5	in	in	ADP
cana-5767	8	6	diverse	diverse	ADJ
cana-5767	8	7	areas	area	NOUN
cana-5767	8	8	such	such	ADJ
cana-5767	8	9	as	as	ADP
cana-5767	8	10	clustering	clustering	NOUN
cana-5767	8	11	,	,	PUNCT
cana-5767	8	12	dimensionality	dimensionality	NOUN
cana-5767	8	13	reduction	reduction	NOUN
cana-5767	8	14	,	,	PUNCT
cana-5767	8	15	and	and	CCONJ
cana-5767	8	16	kernel	kernel	PROPN
cana-5767	8	17	methods	method	NOUN
cana-5767	8	18	,	,	PUNCT
cana-5767	8	19	where	where	SCONJ
cana-5767	8	20	the	the	DET
cana-5767	8	21	spectral	spectral	ADJ
cana-5767	8	22	properties	property	NOUN
cana-5767	8	23	of	of	ADP
cana-5767	8	24	matrices	matrix	NOUN
cana-5767	8	25	such	such	ADJ
cana-5767	8	26	as	as	ADP
cana-5767	8	27	the	the	DET
cana-5767	8	28	graph	graph	NOUN
cana-5767	8	29	laplacian	laplacian	PROPN
cana-5767	8	30	,	,	PUNCT
cana-5767	8	31	covariance	covariance	NOUN
cana-5767	8	32	matrices	matrix	NOUN
cana-5767	8	33	,	,	PUNCT
cana-5767	8	34	and	and	CCONJ
cana-5767	8	35	kernel	kernel	PROPN
cana-5767	8	36	matrices	matrix	NOUN
cana-5767	8	37	play	play	VERB
cana-5767	8	38	a	a	DET
cana-5767	8	39	central	central	ADJ
cana-5767	8	40	role	role	NOUN
cana-5767	8	41	.	.	PUNCT
cana-5767	9	1	for	for	ADP
cana-5767	9	2	instance	instance	NOUN
cana-5767	9	3	,	,	PUNCT
cana-5767	9	4	in	in	ADP
cana-5767	9	5	spectral	spectral	ADJ
cana-5767	9	6	clustering	clustering	NOUN
cana-5767	9	7	,	,	PUNCT
cana-5767	9	8	the	the	DET
cana-5767	9	9	eigenvalues	eigenvalue	NOUN
cana-5767	9	10	and	and	CCONJ
cana-5767	9	11	eigenvectors	eigenvector	NOUN
cana-5767	9	12	of	of	ADP
cana-5767	9	13	the	the	DET
cana-5767	9	14	graph	graph	NOUN
cana-5767	9	15	laplacian	laplacian	NOUN
cana-5767	9	16	are	be	AUX
cana-5767	9	17	used	use	VERB
cana-5767	9	18	to	to	PART
cana-5767	9	19	partition	partition	VERB
cana-5767	9	20	datasets	dataset	NOUN
cana-5767	9	21	into	into	ADP
cana-5767	9	22	clusters	cluster	NOUN
cana-5767	9	23	,	,	PUNCT
cana-5767	9	24	while	while	SCONJ
cana-5767	9	25	in	in	ADP
cana-5767	9	26	kernel	kernel	PROPN
cana-5767	9	27	principal	principal	ADJ
cana-5767	9	28	component	component	NOUN
cana-5767	9	29	analysis	analysis	NOUN
cana-5767	9	30	(	(	PUNCT
cana-5767	9	31	kpca	kpca	PROPN
cana-5767	9	32	)	)	PUNCT
cana-5767	9	33	,	,	PUNCT
cana-5767	9	34	the	the	DET
cana-5767	9	35	eigendecomposition	eigendecomposition	NOUN
cana-5767	9	36	of	of	ADP
cana-5767	9	37	the	the	DET
cana-5767	9	38	kernel	kernel	NOUN
cana-5767	9	39	matrix	matrix	NOUN
cana-5767	9	40	allows	allow	VERB
cana-5767	9	41	for	for	ADP
cana-5767	9	42	non	non	ADJ
cana-5767	9	43	-	-	ADJ
cana-5767	9	44	linear	linear	ADJ
cana-5767	9	45	dimensionality	dimensionality	NOUN
cana-5767	9	46	reduction	reduction	NOUN
cana-5767	9	47	.	.	PUNCT
cana-5767	10	1	despite	despite	SCONJ
cana-5767	10	2	their	their	PRON
cana-5767	10	3	strengths	strength	NOUN
cana-5767	10	4	,	,	PUNCT
cana-5767	10	5	spectral	spectral	ADJ
cana-5767	10	6	methods	method	NOUN
cana-5767	10	7	are	be	AUX
cana-5767	10	8	not	not	PART
cana-5767	10	9	immune	immune	ADJ
cana-5767	10	10	to	to	ADP
cana-5767	10	11	challenges	challenge	NOUN
cana-5767	10	12	,	,	PUNCT
cana-5767	10	13	particularly	particularly	ADV
cana-5767	10	14	when	when	SCONJ
cana-5767	10	15	the	the	DET
cana-5767	10	16	input	input	NOUN
cana-5767	10	17	data	data	NOUN
cana-5767	10	18	is	be	AUX
cana-5767	10	19	subject	subject	ADJ
cana-5767	10	20	to	to	ADP
cana-5767	10	21	noise	noise	NOUN
cana-5767	10	22	,	,	PUNCT
cana-5767	10	23	small	small	ADJ
cana-5767	10	24	perturbations	perturbation	NOUN
cana-5767	10	25	,	,	PUNCT
cana-5767	10	26	or	or	CCONJ
cana-5767	10	27	adversarial	adversarial	ADJ
cana-5767	10	28	attacks	attack	NOUN
cana-5767	10	29	.	.	PUNCT
cana-5767	11	1	in	in	ADP
cana-5767	11	2	adversarial	adversarial	ADJ
cana-5767	11	3	settings	setting	NOUN
cana-5767	11	4	,	,	PUNCT
cana-5767	11	5	even	even	ADV
cana-5767	11	6	minor	minor	ADJ
cana-5767	11	7	perturbations	perturbation	NOUN
cana-5767	11	8	to	to	ADP
cana-5767	11	9	the	the	DET
cana-5767	11	10	input	input	NOUN
cana-5767	11	11	data	datum	NOUN
cana-5767	11	12	can	can	AUX
cana-5767	11	13	cause	cause	VERB
cana-5767	11	14	significant	significant	ADJ
cana-5767	11	15	shifts	shift	NOUN
cana-5767	11	16	in	in	ADP
cana-5767	11	17	the	the	DET
cana-5767	11	18	spectral	spectral	ADJ
cana-5767	11	19	properties	property	NOUN
cana-5767	11	20	of	of	ADP
cana-5767	11	21	matrices	matrix	NOUN
cana-5767	11	22	,	,	PUNCT
cana-5767	11	23	leading	lead	VERB
cana-5767	11	24	to	to	ADP
cana-5767	11	25	instability	instability	NOUN
cana-5767	11	26	in	in	ADP
cana-5767	11	27	the	the	DET
cana-5767	11	28	results	result	NOUN
cana-5767	11	29	produced	produce	VERB
cana-5767	11	30	by	by	ADP
cana-5767	11	31	spectral	spectral	ADJ
cana-5767	11	32	algorithms	algorithm	NOUN
cana-5767	11	33	.	.	PUNCT
cana-5767	12	1	for	for	ADP
cana-5767	12	2	example	example	NOUN
cana-5767	12	3	,	,	PUNCT
cana-5767	12	4	in	in	ADP
cana-5767	12	5	graph	graph	NOUN
cana-5767	12	6	-	-	PUNCT
cana-5767	12	7	based	base	VERB
cana-5767	12	8	learning	learning	NOUN
cana-5767	12	9	,	,	PUNCT
cana-5767	12	10	small	small	ADJ
cana-5767	12	11	changes	change	NOUN
cana-5767	12	12	in	in	ADP
cana-5767	12	13	the	the	DET
cana-5767	12	14	graph	graph	NOUN
cana-5767	12	15	’s	’s	PART
cana-5767	12	16	edge	edge	NOUN
cana-5767	12	17	weights	weight	NOUN
cana-5767	12	18	or	or	CCONJ
cana-5767	12	19	structure	structure	NOUN
cana-5767	12	20	may	may	AUX
cana-5767	12	21	drastically	drastically	ADV
cana-5767	12	22	alter	alter	VERB
cana-5767	12	23	the	the	DET
cana-5767	12	24	eigenvalues	eigenvalue	NOUN
cana-5767	12	25	and	and	CCONJ
cana-5767	12	26	eigenvectors	eigenvector	NOUN
cana-5767	12	27	of	of	ADP
cana-5767	12	28	the	the	DET
cana-5767	12	29	graph	graph	NOUN
cana-5767	12	30	laplacian	laplacian	NOUN
cana-5767	12	31	,	,	PUNCT
cana-5767	12	32	potentially	potentially	ADV
cana-5767	12	33	impacting	impact	VERB
cana-5767	12	34	downstream	downstream	ADJ
cana-5767	12	35	tasks	task	NOUN
cana-5767	12	36	like	like	ADP
cana-5767	12	37	clustering	clustering	NOUN
cana-5767	12	38	or	or	CCONJ
cana-5767	12	39	classification	classification	NOUN
cana-5767	12	40	.	.	PUNCT
cana-5767	13	1	similarly	similarly	ADV
cana-5767	13	2	,	,	PUNCT
cana-5767	13	3	kernelbased	kernelbase	VERB
cana-5767	13	4	methods	method	NOUN
cana-5767	13	5	can	can	AUX
cana-5767	13	6	be	be	AUX
cana-5767	13	7	sensitive	sensitive	ADJ
cana-5767	13	8	to	to	ADP
cana-5767	13	9	perturbations	perturbation	NOUN
cana-5767	13	10	in	in	ADP
cana-5767	13	11	the	the	DET
cana-5767	13	12	input	input	NOUN
cana-5767	13	13	space	space	NOUN
cana-5767	13	14	,	,	PUNCT
cana-5767	13	15	as	as	SCONJ
cana-5767	13	16	these	these	PRON
cana-5767	13	17	directly	directly	ADV
cana-5767	13	18	influence	influence	VERB
cana-5767	13	19	the	the	DET
cana-5767	13	20	kernel	kernel	PROPN
cana-5767	13	21	matrix	matrix	NOUN
cana-5767	13	22	,	,	PUNCT
cana-5767	13	23	altering	alter	VERB
cana-5767	13	24	its	its	PRON
cana-5767	13	25	spectral	spectral	ADJ
cana-5767	13	26	decomposition	decomposition	NOUN
cana-5767	13	27	.	.	PUNCT
cana-5767	14	1	this	this	DET
cana-5767	14	2	vulnerability	vulnerability	NOUN
cana-5767	14	3	raises	raise	VERB
cana-5767	14	4	concerns	concern	NOUN
cana-5767	14	5	about	about	ADP
cana-5767	14	6	mailto:1082usha@gmail.com	mailto:1082usha@gmail.com	PROPN
cana-5767	14	7	communications	communication	NOUN
cana-5767	14	8	on	on	ADP
cana-5767	14	9	applied	apply	VERB
cana-5767	14	10	nonlinear	nonlinear	ADJ
cana-5767	14	11	analysis	analysis	NOUN
cana-5767	14	12	issn	issn	NOUN
cana-5767	14	13	:	:	PUNCT
cana-5767	14	14	1074	1074	NUM
cana-5767	14	15	-	-	PUNCT
cana-5767	14	16	133x	133x	NUM
cana-5767	14	17	vol	vol	NOUN
cana-5767	14	18	32	32	NUM
cana-5767	14	19	no	no	NOUN
cana-5767	14	20	.	.	NOUN
cana-5767	14	21	1	1	NUM
cana-5767	14	22	(	(	PUNCT
cana-5767	14	23	2025	2025	NUM
cana-5767	14	24	)	)	PUNCT
cana-5767	14	25	560	560	NUM
cana-5767	14	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-5767	14	27	the	the	DET
cana-5767	14	28	robustness	robustness	NOUN
cana-5767	14	29	and	and	CCONJ
cana-5767	14	30	reliability	reliability	NOUN
cana-5767	14	31	of	of	ADP
cana-5767	14	32	spectral	spectral	ADJ
cana-5767	14	33	methods	method	NOUN
cana-5767	14	34	,	,	PUNCT
cana-5767	14	35	particularly	particularly	ADV
cana-5767	14	36	in	in	ADP
cana-5767	14	37	real	real	ADJ
cana-5767	14	38	-	-	PUNCT
cana-5767	14	39	world	world	NOUN
cana-5767	14	40	applications	application	NOUN
cana-5767	14	41	where	where	SCONJ
cana-5767	14	42	noisy	noisy	ADJ
cana-5767	14	43	or	or	CCONJ
cana-5767	14	44	adversarial	adversarial	ADJ
cana-5767	14	45	data	datum	NOUN
cana-5767	14	46	is	be	AUX
cana-5767	14	47	common	common	ADJ
cana-5767	14	48	.	.	PUNCT
cana-5767	15	1	to	to	PART
cana-5767	15	2	address	address	VERB
cana-5767	15	3	these	these	DET
cana-5767	15	4	challenges	challenge	NOUN
cana-5767	15	5	,	,	PUNCT
cana-5767	15	6	we	we	PRON
cana-5767	15	7	turn	turn	VERB
cana-5767	15	8	to	to	ADP
cana-5767	15	9	operator	operator	NOUN
cana-5767	15	10	theory	theory	NOUN
cana-5767	15	11	,	,	PUNCT
cana-5767	15	12	specifically	specifically	ADV
cana-5767	15	13	leveraging	leverage	VERB
cana-5767	15	14	weyl	weyl	NOUN
cana-5767	15	15	’s	’s	PART
cana-5767	15	16	theorem	theorem	NOUN
cana-5767	15	17	on	on	ADP
cana-5767	15	18	the	the	DET
cana-5767	15	19	stability	stability	NOUN
cana-5767	15	20	of	of	ADP
cana-5767	15	21	the	the	DET
cana-5767	15	22	essential	essential	ADJ
cana-5767	15	23	spectrum	spectrum	NOUN
cana-5767	15	24	of	of	ADP
cana-5767	15	25	operators	operator	NOUN
cana-5767	15	26	under	under	ADP
cana-5767	15	27	compact	compact	ADJ
cana-5767	15	28	perturbations	perturbation	NOUN
cana-5767	15	29	.	.	PUNCT
cana-5767	16	1	weyl	weyl	PROPN
cana-5767	16	2	’s	’s	PART
cana-5767	16	3	theorem	theorem	NOUN
cana-5767	16	4	provides	provide	VERB
cana-5767	16	5	a	a	DET
cana-5767	16	6	powerful	powerful	ADJ
cana-5767	16	7	framework	framework	NOUN
cana-5767	16	8	for	for	ADP
cana-5767	16	9	understanding	understand	VERB
cana-5767	16	10	how	how	SCONJ
cana-5767	16	11	the	the	DET
cana-5767	16	12	eigenvalues	eigenvalue	NOUN
cana-5767	16	13	of	of	ADP
cana-5767	16	14	an	an	DET
cana-5767	16	15	operator	operator	NOUN
cana-5767	16	16	change	change	NOUN
cana-5767	16	17	when	when	SCONJ
cana-5767	16	18	subjected	subject	VERB
cana-5767	16	19	to	to	ADP
cana-5767	16	20	small	small	ADJ
cana-5767	16	21	perturbations	perturbation	NOUN
cana-5767	16	22	.	.	PUNCT
cana-5767	17	1	in	in	ADP
cana-5767	17	2	the	the	DET
cana-5767	17	3	context	context	NOUN
cana-5767	17	4	of	of	ADP
cana-5767	17	5	machine	machine	NOUN
cana-5767	17	6	learning	learning	NOUN
cana-5767	17	7	,	,	PUNCT
cana-5767	17	8	this	this	PRON
cana-5767	17	9	implies	imply	VERB
cana-5767	17	10	that	that	SCONJ
cana-5767	17	11	while	while	SCONJ
cana-5767	17	12	individual	individual	ADJ
cana-5767	17	13	eigenvalues	eigenvalue	NOUN
cana-5767	17	14	may	may	AUX
cana-5767	17	15	shift	shift	VERB
cana-5767	17	16	under	under	ADP
cana-5767	17	17	data	datum	NOUN
cana-5767	17	18	perturbations	perturbation	NOUN
cana-5767	17	19	,	,	PUNCT
cana-5767	17	20	the	the	DET
cana-5767	17	21	essential	essential	ADJ
cana-5767	17	22	spectrum	spectrum	NOUN
cana-5767	17	23	—	—	PUNCT
cana-5767	17	24	which	which	PRON
cana-5767	17	25	governs	govern	VERB
cana-5767	17	26	the	the	DET
cana-5767	17	27	global	global	ADJ
cana-5767	17	28	structure	structure	NOUN
cana-5767	17	29	—	—	PUNCT
cana-5767	17	30	remains	remain	VERB
cana-5767	17	31	stable	stable	ADJ
cana-5767	17	32	.	.	PUNCT
cana-5767	18	1	this	this	DET
cana-5767	18	2	stability	stability	NOUN
cana-5767	18	3	is	be	AUX
cana-5767	18	4	crucial	crucial	ADJ
cana-5767	18	5	for	for	ADP
cana-5767	18	6	ensuring	ensure	VERB
cana-5767	18	7	that	that	SCONJ
cana-5767	18	8	the	the	DET
cana-5767	18	9	overall	overall	ADJ
cana-5767	18	10	structure	structure	NOUN
cana-5767	18	11	captured	capture	VERB
cana-5767	18	12	by	by	ADP
cana-5767	18	13	spectral	spectral	ADJ
cana-5767	18	14	methods	method	NOUN
cana-5767	18	15	is	be	AUX
cana-5767	18	16	robust	robust	ADJ
cana-5767	18	17	,	,	PUNCT
cana-5767	18	18	even	even	ADV
cana-5767	18	19	in	in	ADP
cana-5767	18	20	the	the	DET
cana-5767	18	21	presence	presence	NOUN
cana-5767	18	22	of	of	ADP
cana-5767	18	23	adversarial	adversarial	ADJ
cana-5767	18	24	or	or	CCONJ
cana-5767	18	25	noisy	noisy	ADJ
cana-5767	18	26	data	datum	NOUN
cana-5767	18	27	.	.	PUNCT
cana-5767	19	1	in	in	ADP
cana-5767	19	2	this	this	DET
cana-5767	19	3	paper	paper	NOUN
cana-5767	19	4	,	,	PUNCT
cana-5767	19	5	we	we	PRON
cana-5767	19	6	investigate	investigate	VERB
cana-5767	19	7	the	the	DET
cana-5767	19	8	application	application	NOUN
cana-5767	19	9	of	of	ADP
cana-5767	19	10	weyl	weyl	PROPN
cana-5767	19	11	’s	’s	PART
cana-5767	19	12	theorem	theorem	NOUN
cana-5767	19	13	to	to	ADP
cana-5767	19	14	spectral	spectral	ADJ
cana-5767	19	15	methods	method	NOUN
cana-5767	19	16	commonly	commonly	ADV
cana-5767	19	17	used	use	VERB
cana-5767	19	18	in	in	ADP
cana-5767	19	19	machine	machine	NOUN
cana-5767	19	20	learning	learning	NOUN
cana-5767	19	21	.	.	PUNCT
cana-5767	20	1	our	our	PRON
cana-5767	20	2	focus	focus	NOUN
cana-5767	20	3	lies	lie	VERB
cana-5767	20	4	on	on	ADP
cana-5767	20	5	two	two	NUM
cana-5767	20	6	key	key	ADJ
cana-5767	20	7	operators	operator	NOUN
cana-5767	20	8	1	1	NUM
cana-5767	20	9	.	.	PUNCT
cana-5767	21	1	the	the	DET
cana-5767	21	2	graph	graph	NOUN
cana-5767	21	3	laplacian	laplacian	NOUN
cana-5767	21	4	:	:	PUNCT
cana-5767	21	5	widely	widely	ADV
cana-5767	21	6	used	use	VERB
cana-5767	21	7	in	in	ADP
cana-5767	21	8	spectral	spectral	ADJ
cana-5767	21	9	clustering	clustering	NOUN
cana-5767	21	10	and	and	CCONJ
cana-5767	21	11	graph	graph	NOUN
cana-5767	21	12	-	-	PUNCT
cana-5767	21	13	based	base	VERB
cana-5767	21	14	learning	learning	NOUN
cana-5767	21	15	,	,	PUNCT
cana-5767	21	16	its	its	PRON
cana-5767	21	17	spectral	spectral	ADJ
cana-5767	21	18	properties	property	NOUN
cana-5767	21	19	determine	determine	VERB
cana-5767	21	20	the	the	DET
cana-5767	21	21	community	community	NOUN
cana-5767	21	22	structure	structure	NOUN
cana-5767	21	23	of	of	ADP
cana-5767	21	24	graphs	graph	NOUN
cana-5767	21	25	and	and	CCONJ
cana-5767	21	26	influence	influence	NOUN
cana-5767	21	27	tasks	task	NOUN
cana-5767	21	28	like	like	ADP
cana-5767	21	29	node	node	NOUN
cana-5767	21	30	classification	classification	NOUN
cana-5767	21	31	and	and	CCONJ
cana-5767	21	32	link	link	NOUN
cana-5767	21	33	prediction	prediction	NOUN
cana-5767	21	34	.	.	PUNCT
cana-5767	22	1	2	2	X
cana-5767	22	2	.	.	X
cana-5767	22	3	kernel	kernel	PROPN
cana-5767	22	4	matrices	matrix	NOUN
cana-5767	22	5	:	:	PUNCT
cana-5767	22	6	central	central	ADJ
cana-5767	22	7	to	to	ADP
cana-5767	22	8	kernel	kernel	PROPN
cana-5767	22	9	methods	method	NOUN
cana-5767	22	10	,	,	PUNCT
cana-5767	22	11	these	these	DET
cana-5767	22	12	matrices	matrix	NOUN
cana-5767	22	13	define	define	VERB
cana-5767	22	14	the	the	DET
cana-5767	22	15	feature	feature	NOUN
cana-5767	22	16	space	space	NOUN
cana-5767	22	17	for	for	ADP
cana-5767	22	18	non	non	ADJ
cana-5767	22	19	-	-	ADJ
cana-5767	22	20	linear	linear	ADJ
cana-5767	22	21	learning	learning	NOUN
cana-5767	22	22	algorithms	algorithm	NOUN
cana-5767	22	23	such	such	ADJ
cana-5767	22	24	as	as	ADP
cana-5767	22	25	support	support	NOUN
cana-5767	22	26	vector	vector	NOUN
cana-5767	22	27	machines	machine	NOUN
cana-5767	22	28	(	(	PUNCT
cana-5767	22	29	svm	svm	PROPN
cana-5767	22	30	)	)	PUNCT
cana-5767	22	31	and	and	CCONJ
cana-5767	22	32	kpca	kpca	NOUN
cana-5767	22	33	.	.	PUNCT
cana-5767	23	1	we	we	PRON
cana-5767	23	2	show	show	VERB
cana-5767	23	3	that	that	SCONJ
cana-5767	23	4	the	the	DET
cana-5767	23	5	stability	stability	NOUN
cana-5767	23	6	of	of	ADP
cana-5767	23	7	the	the	DET
cana-5767	23	8	essential	essential	ADJ
cana-5767	23	9	spectrum	spectrum	NOUN
cana-5767	23	10	under	under	ADP
cana-5767	23	11	compact	compact	ADJ
cana-5767	23	12	perturbations	perturbation	NOUN
cana-5767	23	13	ensures	ensure	VERB
cana-5767	23	14	robustness	robustness	NOUN
cana-5767	23	15	in	in	ADP
cana-5767	23	16	these	these	DET
cana-5767	23	17	spectral	spectral	ADJ
cana-5767	23	18	methods	method	NOUN
cana-5767	23	19	,	,	PUNCT
cana-5767	23	20	mitigating	mitigate	VERB
cana-5767	23	21	the	the	DET
cana-5767	23	22	impact	impact	NOUN
cana-5767	23	23	of	of	ADP
cana-5767	23	24	small	small	ADJ
cana-5767	23	25	adversarial	adversarial	ADJ
cana-5767	23	26	changes	change	NOUN
cana-5767	23	27	to	to	ADP
cana-5767	23	28	the	the	DET
cana-5767	23	29	input	input	NOUN
cana-5767	23	30	data	datum	NOUN
cana-5767	23	31	.	.	PUNCT
cana-5767	24	1	through	through	ADP
cana-5767	24	2	a	a	DET
cana-5767	24	3	combination	combination	NOUN
cana-5767	24	4	of	of	ADP
cana-5767	24	5	theoretical	theoretical	ADJ
cana-5767	24	6	analysis	analysis	NOUN
cana-5767	24	7	,	,	PUNCT
cana-5767	24	8	proofs	proof	NOUN
cana-5767	24	9	,	,	PUNCT
cana-5767	24	10	and	and	CCONJ
cana-5767	24	11	practical	practical	ADJ
cana-5767	24	12	experiments	experiment	NOUN
cana-5767	24	13	,	,	PUNCT
cana-5767	24	14	we	we	PRON
cana-5767	24	15	aim	aim	VERB
cana-5767	24	16	to	to	PART
cana-5767	24	17	bridge	bridge	VERB
cana-5767	24	18	the	the	DET
cana-5767	24	19	gap	gap	NOUN
cana-5767	24	20	between	between	ADP
cana-5767	24	21	operator	operator	NOUN
cana-5767	24	22	theory	theory	NOUN
cana-5767	24	23	and	and	CCONJ
cana-5767	24	24	machine	machine	NOUN
cana-5767	24	25	learning	learning	NOUN
cana-5767	24	26	,	,	PUNCT
cana-5767	24	27	providing	provide	VERB
cana-5767	24	28	insights	insight	NOUN
cana-5767	24	29	into	into	ADP
cana-5767	24	30	how	how	SCONJ
cana-5767	24	31	mathematical	mathematical	ADJ
cana-5767	24	32	stability	stability	NOUN
cana-5767	24	33	translates	translate	VERB
cana-5767	24	34	into	into	ADP
cana-5767	24	35	robust	robust	ADJ
cana-5767	24	36	algorithms	algorithm	NOUN
cana-5767	24	37	.	.	PUNCT
cana-5767	25	1	the	the	DET
cana-5767	25	2	main	main	ADJ
cana-5767	25	3	contributions	contribution	NOUN
cana-5767	25	4	of	of	ADP
cana-5767	25	5	this	this	DET
cana-5767	25	6	paper	paper	NOUN
cana-5767	25	7	are	be	AUX
cana-5767	25	8	as	as	SCONJ
cana-5767	25	9	follows	follow	VERB
cana-5767	25	10	:	:	PUNCT
cana-5767	25	11	1	1	X
cana-5767	25	12	.	.	X
cana-5767	25	13	theoretical	theoretical	ADJ
cana-5767	25	14	foundation	foundation	NOUN
cana-5767	25	15	:	:	PUNCT
cana-5767	25	16	we	we	PRON
cana-5767	25	17	extend	extend	VERB
cana-5767	25	18	weyl	weyl	PROPN
cana-5767	25	19	’s	’s	PART
cana-5767	25	20	theorem	theorem	NOUN
cana-5767	25	21	to	to	ADP
cana-5767	25	22	specific	specific	ADJ
cana-5767	25	23	operators	operator	NOUN
cana-5767	25	24	used	use	VERB
cana-5767	25	25	in	in	ADP
cana-5767	25	26	spectral	spectral	ADJ
cana-5767	25	27	methods	method	NOUN
cana-5767	25	28	,	,	PUNCT
cana-5767	25	29	demonstrating	demonstrate	VERB
cana-5767	25	30	its	its	PRON
cana-5767	25	31	relevance	relevance	NOUN
cana-5767	25	32	to	to	PART
cana-5767	25	33	graph	graph	VERB
cana-5767	25	34	laplacians	laplacian	NOUN
cana-5767	25	35	and	and	CCONJ
cana-5767	25	36	kernel	kernel	PROPN
cana-5767	25	37	matrices	matrix	NOUN
cana-5767	25	38	.	.	PUNCT
cana-5767	26	1	2	2	X
cana-5767	26	2	.	.	X
cana-5767	26	3	adversarial	adversarial	ADJ
cana-5767	26	4	robustness	robustness	NOUN
cana-5767	26	5	:	:	PUNCT
cana-5767	26	6	we	we	PRON
cana-5767	26	7	analyse	analyse	VERB
cana-5767	26	8	how	how	SCONJ
cana-5767	26	9	stability	stability	NOUN
cana-5767	26	10	in	in	ADP
cana-5767	26	11	the	the	DET
cana-5767	26	12	essential	essential	ADJ
cana-5767	26	13	spectrum	spectrum	NOUN
cana-5767	26	14	impacts	impact	VERB
cana-5767	26	15	the	the	DET
cana-5767	26	16	resilience	resilience	NOUN
cana-5767	26	17	of	of	ADP
cana-5767	26	18	spectral	spectral	ADJ
cana-5767	26	19	methods	method	NOUN
cana-5767	26	20	to	to	ADP
cana-5767	26	21	adversarial	adversarial	ADJ
cana-5767	26	22	perturbations	perturbation	NOUN
cana-5767	26	23	and	and	CCONJ
cana-5767	26	24	noise	noise	NOUN
cana-5767	26	25	.	.	PUNCT
cana-5767	27	1	3	3	X
cana-5767	27	2	.	.	X
cana-5767	27	3	numerical	numerical	ADJ
cana-5767	27	4	experiments	experiment	NOUN
cana-5767	27	5	:	:	PUNCT
cana-5767	27	6	we	we	PRON
cana-5767	27	7	implement	implement	VERB
cana-5767	27	8	spectral	spectral	ADJ
cana-5767	27	9	clustering	clustering	NOUN
cana-5767	27	10	and	and	CCONJ
cana-5767	27	11	kernel	kernel	NOUN
cana-5767	27	12	-	-	PUNCT
cana-5767	27	13	based	base	VERB
cana-5767	27	14	methods	method	NOUN
cana-5767	27	15	on	on	ADP
cana-5767	27	16	perturbed	perturb	VERB
cana-5767	27	17	datasets	dataset	NOUN
cana-5767	27	18	,	,	PUNCT
cana-5767	27	19	illustrating	illustrate	VERB
cana-5767	27	20	how	how	SCONJ
cana-5767	27	21	the	the	DET
cana-5767	27	22	stability	stability	NOUN
cana-5767	27	23	results	result	VERB
cana-5767	27	24	translate	translate	VERB
cana-5767	27	25	to	to	ADP
cana-5767	27	26	practical	practical	ADJ
cana-5767	27	27	robustness	robustness	NOUN
cana-5767	27	28	.	.	PUNCT
cana-5767	28	1	4	4	X
cana-5767	28	2	.	.	X
cana-5767	28	3	applications	application	NOUN
cana-5767	28	4	:	:	PUNCT
cana-5767	28	5	we	we	PRON
cana-5767	28	6	highlight	highlight	VERB
cana-5767	28	7	the	the	DET
cana-5767	28	8	implications	implication	NOUN
cana-5767	28	9	of	of	ADP
cana-5767	28	10	our	our	PRON
cana-5767	28	11	findings	finding	NOUN
cana-5767	28	12	for	for	ADP
cana-5767	28	13	real	real	ADJ
cana-5767	28	14	-	-	PUNCT
cana-5767	28	15	world	world	NOUN
cana-5767	28	16	problems	problem	NOUN
cana-5767	28	17	such	such	ADJ
cana-5767	28	18	as	as	ADP
cana-5767	28	19	graph	graph	NOUN
cana-5767	28	20	-	-	PUNCT
cana-5767	28	21	based	base	VERB
cana-5767	28	22	learning	learning	NOUN
cana-5767	28	23	,	,	PUNCT
cana-5767	28	24	clustering	clustering	NOUN
cana-5767	28	25	,	,	PUNCT
cana-5767	28	26	and	and	CCONJ
cana-5767	28	27	dimensionality	dimensionality	NOUN
cana-5767	28	28	reduction	reduction	NOUN
cana-5767	28	29	in	in	ADP
cana-5767	28	30	adversarial	adversarial	ADJ
cana-5767	28	31	or	or	CCONJ
cana-5767	28	32	noisy	noisy	ADJ
cana-5767	28	33	environments	environment	NOUN
cana-5767	28	34	.	.	PUNCT
cana-5767	29	1	by	by	ADP
cana-5767	29	2	combining	combine	VERB
cana-5767	29	3	operator	operator	NOUN
cana-5767	29	4	theory	theory	NOUN
cana-5767	29	5	with	with	ADP
cana-5767	29	6	practical	practical	ADJ
cana-5767	29	7	insights	insight	NOUN
cana-5767	29	8	into	into	ADP
cana-5767	29	9	machine	machine	NOUN
cana-5767	29	10	learning	learning	NOUN
cana-5767	29	11	,	,	PUNCT
cana-5767	29	12	this	this	DET
cana-5767	29	13	work	work	NOUN
cana-5767	29	14	not	not	PART
cana-5767	29	15	only	only	ADV
cana-5767	29	16	deepens	deepen	VERB
cana-5767	29	17	our	our	PRON
cana-5767	29	18	understanding	understanding	NOUN
cana-5767	29	19	of	of	ADP
cana-5767	29	20	spectral	spectral	ADJ
cana-5767	29	21	stability	stability	NOUN
cana-5767	29	22	but	but	CCONJ
cana-5767	29	23	also	also	ADV
cana-5767	29	24	contributes	contribute	VERB
cana-5767	29	25	to	to	ADP
cana-5767	29	26	the	the	DET
cana-5767	29	27	development	development	NOUN
cana-5767	29	28	of	of	ADP
cana-5767	29	29	more	more	ADV
cana-5767	29	30	robust	robust	ADJ
cana-5767	29	31	and	and	CCONJ
cana-5767	29	32	reliable	reliable	ADJ
cana-5767	29	33	spectral	spectral	NOUN
cana-5767	29	34	-	-	PUNCT
cana-5767	29	35	based	base	VERB
cana-5767	29	36	algorithms	algorithm	NOUN
cana-5767	29	37	.	.	PUNCT
cana-5767	30	1	communications	communication	NOUN
cana-5767	30	2	on	on	ADP
cana-5767	30	3	applied	apply	VERB
cana-5767	30	4	nonlinear	nonlinear	ADJ
cana-5767	30	5	analysis	analysis	NOUN
cana-5767	30	6	issn	issn	NOUN
cana-5767	30	7	:	:	PUNCT
cana-5767	30	8	1074	1074	NUM
cana-5767	30	9	-	-	PUNCT
cana-5767	30	10	133x	133x	NUM
cana-5767	30	11	vol	vol	NOUN
cana-5767	30	12	32	32	NUM
cana-5767	30	13	no	no	NOUN
cana-5767	30	14	.	.	NOUN
cana-5767	30	15	1	1	NUM
cana-5767	30	16	(	(	PUNCT
cana-5767	30	17	2025	2025	NUM
cana-5767	30	18	)	)	PUNCT
cana-5767	30	19	561	561	NUM
cana-5767	30	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-5767	30	21	stability	stability	NOUN
cana-5767	30	22	of	of	ADP
cana-5767	30	23	essential	essential	ADJ
cana-5767	30	24	spectrum	spectrum	NOUN
cana-5767	30	25	under	under	ADP
cana-5767	30	26	perturbations	perturbation	NOUN
cana-5767	30	27	theorem	theorem	VERB
cana-5767	30	28	1	1	NUM
cana-5767	30	29	(	(	PUNCT
cana-5767	30	30	weyl	weyl	PROPN
cana-5767	30	31	’s	’s	PART
cana-5767	30	32	theorem	theorem	NOUN
cana-5767	30	33	for	for	ADP
cana-5767	30	34	spectral	spectral	ADJ
cana-5767	30	35	stability	stability	NOUN
cana-5767	30	36	):	):	PUNCT
cana-5767	30	37	let	let	VERB
cana-5767	30	38	l	l	NOUN
cana-5767	30	39	be	be	AUX
cana-5767	30	40	a	a	DET
cana-5767	30	41	self	self	NOUN
cana-5767	30	42	-	-	PUNCT
cana-5767	30	43	adjoint	adjoint	NOUN
cana-5767	30	44	operator	operator	NOUN
cana-5767	30	45	corresponding	correspond	VERB
cana-5767	30	46	to	to	ADP
cana-5767	30	47	the	the	DET
cana-5767	30	48	graph	graph	NOUN
cana-5767	30	49	laplacian	laplacian	ADJ
cana-5767	30	50	or	or	CCONJ
cana-5767	30	51	kernel	kernel	NOUN
cana-5767	30	52	matrix	matrix	NOUN
cana-5767	30	53	,	,	PUNCT
cana-5767	30	54	and	and	CCONJ
cana-5767	30	55	let	let	VERB
cana-5767	30	56	p	p	PRON
cana-5767	30	57	be	be	AUX
cana-5767	30	58	a	a	DET
cana-5767	30	59	small	small	ADJ
cana-5767	30	60	perturbation	perturbation	NOUN
cana-5767	30	61	.	.	PUNCT
cana-5767	31	1	then	then	ADV
cana-5767	31	2	,	,	PUNCT
cana-5767	31	3	for	for	ADP
cana-5767	31	4	sufficiently	sufficiently	ADV
cana-5767	31	5	small	small	ADJ
cana-5767	31	6	p	p	NOUN
cana-5767	31	7	,	,	PUNCT
cana-5767	31	8	the	the	DET
cana-5767	31	9	spectrum	spectrum	NOUN
cana-5767	31	10	of	of	ADP
cana-5767	31	11	the	the	DET
cana-5767	31	12	perturbed	perturb	VERB
cana-5767	31	13	operator	operator	NOUN
cana-5767	31	14	l+pl	l+pl	VERB
cana-5767	32	1	+	+	ADV
cana-5767	32	2	p	p	NOUN
cana-5767	32	3	is	be	AUX
cana-5767	32	4	close	close	ADJ
cana-5767	32	5	to	to	ADP
cana-5767	32	6	the	the	DET
cana-5767	32	7	spectrum	spectrum	NOUN
cana-5767	32	8	of	of	ADP
cana-5767	32	9	l.	l.	PROPN
cana-5767	32	10	specifically	specifically	ADV
cana-5767	32	11	,	,	PUNCT
cana-5767	32	12	the	the	DET
cana-5767	32	13	essential	essential	ADJ
cana-5767	32	14	spectrum	spectrum	NOUN
cana-5767	32	15	of	of	ADP
cana-5767	32	16	l+pl	l+pl	PROPN
cana-5767	32	17	+	+	X
cana-5767	32	18	p	p	NOUN
cana-5767	32	19	remains	remain	VERB
cana-5767	32	20	unchanged	unchanged	ADJ
cana-5767	32	21	:	:	PUNCT
cana-5767	32	22	proof	proof	NOUN
cana-5767	32	23	:	:	PUNCT
cana-5767	32	24	1	1	X
cana-5767	32	25	.	.	X
cana-5767	32	26	compactness	compactness	NOUN
cana-5767	32	27	of	of	ADP
cana-5767	32	28	perturbations	perturbation	NOUN
cana-5767	32	29	:	:	PUNCT
cana-5767	32	30	since	since	SCONJ
cana-5767	32	31	the	the	DET
cana-5767	32	32	perturbation	perturbation	NOUN
cana-5767	32	33	p	p	NOUN
cana-5767	32	34	is	be	AUX
cana-5767	32	35	small	small	ADJ
cana-5767	32	36	,	,	PUNCT
cana-5767	32	37	we	we	PRON
cana-5767	32	38	treat	treat	VERB
cana-5767	32	39	it	it	PRON
cana-5767	32	40	as	as	ADP
cana-5767	32	41	a	a	DET
cana-5767	32	42	compact	compact	ADJ
cana-5767	32	43	operator	operator	NOUN
cana-5767	32	44	(	(	PUNCT
cana-5767	32	45	i.e.	i.e.	X
cana-5767	32	46	,	,	PUNCT
cana-5767	32	47	one	one	NUM
cana-5767	32	48	that	that	PRON
cana-5767	32	49	does	do	AUX
cana-5767	32	50	not	not	PART
cana-5767	32	51	change	change	VERB
cana-5767	32	52	the	the	DET
cana-5767	32	53	global	global	ADJ
cana-5767	32	54	structure	structure	NOUN
cana-5767	32	55	of	of	ADP
cana-5767	32	56	the	the	DET
cana-5767	32	57	graph	graph	NOUN
cana-5767	32	58	significantly	significantly	ADV
cana-5767	32	59	)	)	PUNCT
cana-5767	32	60	.	.	PUNCT
cana-5767	33	1	2	2	X
cana-5767	33	2	.	.	X
cana-5767	33	3	weyl	weyl	PROPN
cana-5767	33	4	's	's	PART
cana-5767	33	5	theorem	theorem	ADJ
cana-5767	33	6	application	application	NOUN
cana-5767	33	7	:	:	PUNCT
cana-5767	33	8	weyl	weyl	PROPN
cana-5767	33	9	’s	’s	PART
cana-5767	33	10	theorem	theorem	ADJ
cana-5767	33	11	guarantees	guarantee	NOUN
cana-5767	33	12	that	that	SCONJ
cana-5767	33	13	the	the	DET
cana-5767	33	14	spectrum	spectrum	NOUN
cana-5767	33	15	of	of	ADP
cana-5767	33	16	the	the	DET
cana-5767	33	17	operator	operator	NOUN
cana-5767	33	18	l+	l+	AUX
cana-5767	33	19	pl+p	pl+p	NUM
cana-5767	33	20	shifts	shift	NOUN
cana-5767	33	21	only	only	ADV
cana-5767	33	22	slightly	slightly	ADV
cana-5767	33	23	.	.	PUNCT
cana-5767	34	1	more	more	ADV
cana-5767	34	2	importantly	importantly	ADV
cana-5767	34	3	,	,	PUNCT
cana-5767	34	4	for	for	ADP
cana-5767	34	5	compact	compact	ADJ
cana-5767	34	6	operators	operator	NOUN
cana-5767	34	7	,	,	PUNCT
cana-5767	34	8	the	the	DET
cana-5767	34	9	essential	essential	ADJ
cana-5767	34	10	spectrum	spectrum	NOUN
cana-5767	34	11	is	be	AUX
cana-5767	34	12	stable	stable	ADJ
cana-5767	34	13	and	and	CCONJ
cana-5767	34	14	does	do	AUX
cana-5767	34	15	not	not	PART
cana-5767	34	16	experience	experience	VERB
cana-5767	34	17	significant	significant	ADJ
cana-5767	34	18	shifts	shift	NOUN
cana-5767	34	19	.	.	PUNCT
cana-5767	35	1	3	3	X
cana-5767	35	2	.	.	X
cana-5767	35	3	conclusion	conclusion	NOUN
cana-5767	35	4	:	:	PUNCT
cana-5767	35	5	hence	hence	ADV
cana-5767	35	6	,	,	PUNCT
cana-5767	35	7	the	the	DET
cana-5767	35	8	essential	essential	ADJ
cana-5767	35	9	spectrum	spectrum	NOUN
cana-5767	35	10	of	of	ADP
cana-5767	35	11	the	the	DET
cana-5767	35	12	perturbed	perturb	VERB
cana-5767	35	13	graph	graph	NOUN
cana-5767	35	14	laplacian	laplacian	ADJ
cana-5767	35	15	(	(	PUNCT
cana-5767	35	16	or	or	CCONJ
cana-5767	35	17	kernel	kernel	PROPN
cana-5767	35	18	matrix	matrix	NOUN
cana-5767	35	19	)	)	PUNCT
cana-5767	35	20	remains	remain	VERB
cana-5767	35	21	equal	equal	ADJ
cana-5767	35	22	to	to	ADP
cana-5767	35	23	that	that	PRON
cana-5767	35	24	of	of	ADP
cana-5767	35	25	the	the	DET
cana-5767	35	26	original	original	ADJ
cana-5767	35	27	operator	operator	NOUN
cana-5767	35	28	.	.	PUNCT
cana-5767	36	1	import	import	NOUN
cana-5767	36	2	numpy	numpy	NOUN
cana-5767	36	3	as	as	ADP
cana-5767	36	4	np	np	NOUN
cana-5767	36	5	from	from	ADP
cana-5767	36	6	sklearn.cluster	sklearn.cluster	PROPN
cana-5767	36	7	import	import	NOUN
cana-5767	36	8	spectralclustering	spectralclustere	VERB
cana-5767	36	9	from	from	ADP
cana-5767	36	10	sklearn.metrics	sklearn.metric	NOUN
cana-5767	36	11	import	import	NOUN
cana-5767	36	12	adjusted_rand_score	adjusted_rand_score	NOUN
cana-5767	36	13	import	import	NOUN
cana-5767	36	14	networkx	networkx	NOUN
cana-5767	36	15	as	as	ADP
cana-5767	36	16	nx	nx	NUM
cana-5767	36	17	import	import	NOUN
cana-5767	36	18	matplotlib.pyplot	matplotlib.pyplot	PROPN
cana-5767	36	19	as	as	SCONJ
cana-5767	36	20	plt	plt	NOUN
cana-5767	36	21	#	#	NOUN
cana-5767	36	22	create	create	VERB
cana-5767	36	23	a	a	DET
cana-5767	36	24	random	random	ADJ
cana-5767	36	25	graph	graph	NOUN
cana-5767	36	26	with	with	ADP
cana-5767	36	27	100	100	NUM
cana-5767	36	28	nodes	node	NOUN
cana-5767	36	29	and	and	CCONJ
cana-5767	36	30	3	3	NUM
cana-5767	36	31	clusters	cluster	NOUN
cana-5767	36	32	n_nodes	n_node	NOUN
cana-5767	36	33	=	=	SYM
cana-5767	36	34	100	100	NUM
cana-5767	36	35	n_clusters	n_cluster	NOUN
cana-5767	36	36	=	=	SYM
cana-5767	36	37	3	3	NUM
cana-5767	36	38	graph	graph	NOUN
cana-5767	36	39	=	=	PUNCT
cana-5767	36	40	nx.erdos_renyi_graph(n_nodes	nx.erdos_renyi_graph(n_node	NOUN
cana-5767	36	41	,	,	PUNCT
cana-5767	36	42	0.1	0.1	NUM
cana-5767	36	43	)	)	PUNCT
cana-5767	36	44	#	#	NOUN
cana-5767	36	45	compute	compute	NOUN
cana-5767	36	46	the	the	DET
cana-5767	36	47	adjacency	adjacency	NOUN
cana-5767	36	48	matrix	matrix	NOUN
cana-5767	36	49	of	of	ADP
cana-5767	36	50	the	the	DET
cana-5767	36	51	graph	graph	NOUN
cana-5767	36	52	adj_matrix	adj_matrix	NOUN
cana-5767	36	53	=	=	SYM
cana-5767	36	54	nx.to_numpy_array(graph	nx.to_numpy_array(graph	PROPN
cana-5767	36	55	)	)	PUNCT
cana-5767	36	56	#	#	NOUN
cana-5767	36	57	apply	apply	VERB
cana-5767	36	58	spectral	spectral	ADJ
cana-5767	36	59	clustering	clustering	NOUN
cana-5767	36	60	to	to	ADP
cana-5767	36	61	the	the	DET
cana-5767	36	62	original	original	ADJ
cana-5767	36	63	adjacency	adjacency	NOUN
cana-5767	36	64	matrix	matrix	NOUN
cana-5767	36	65	sc	sc	NOUN
cana-5767	36	66	=	=	PUNCT
cana-5767	36	67	spectralclustering(n_clusters	spectralclustering(n_cluster	NOUN
cana-5767	36	68	=	=	NOUN
cana-5767	36	69	n_clusters	n_cluster	NOUN
cana-5767	36	70	,	,	PUNCT
cana-5767	36	71	affinity='precomputed	affinity='precompute	VERB
cana-5767	36	72	'	'	NUM
cana-5767	36	73	,	,	PUNCT
cana-5767	36	74	random_state=42	random_state=42	NOUN
cana-5767	36	75	)	)	PUNCT
cana-5767	36	76	labels_original	labels_original	PROPN
cana-5767	36	77	=	=	SYM
cana-5767	36	78	sc.fit_predict(adj_matrix	sc.fit_predict(adj_matrix	PROPN
cana-5767	36	79	)	)	PUNCT
cana-5767	36	80	#	#	NOUN
cana-5767	36	81	add	add	VERB
cana-5767	36	82	random	random	ADJ
cana-5767	36	83	gaussian	gaussian	NOUN
cana-5767	36	84	noise	noise	NOUN
cana-5767	36	85	to	to	ADP
cana-5767	36	86	the	the	DET
cana-5767	36	87	adjacency	adjacency	NOUN
cana-5767	36	88	matrix	matrix	NOUN
cana-5767	36	89	(	(	PUNCT
cana-5767	36	90	perturbation	perturbation	NOUN
cana-5767	36	91	)	)	PUNCT
cana-5767	36	92	noise	noise	NOUN
cana-5767	36	93	=	=	SYM
cana-5767	36	94	np.random.normal(0	np.random.normal(0	PROPN
cana-5767	36	95	,	,	PUNCT
cana-5767	36	96	0.1	0.1	NUM
cana-5767	36	97	,	,	PUNCT
cana-5767	36	98	size	size	NOUN
cana-5767	36	99	=	=	SYM
cana-5767	36	100	adj_matrix.shape	adj_matrix.shape	NOUN
cana-5767	36	101	)	)	PUNCT
cana-5767	36	102	adj_matrix_perturbed	adj_matrix_perturbed	NOUN
cana-5767	36	103	=	=	SYM
cana-5767	36	104	adj_matrix	adj_matrix	NOUN
cana-5767	36	105	+	+	NOUN
cana-5767	36	106	noise	noise	NOUN
cana-5767	36	107	communications	communication	NOUN
cana-5767	36	108	on	on	ADP
cana-5767	36	109	applied	apply	VERB
cana-5767	36	110	nonlinear	nonlinear	ADJ
cana-5767	36	111	analysis	analysis	NOUN
cana-5767	36	112	issn	issn	NOUN
cana-5767	36	113	:	:	PUNCT
cana-5767	36	114	1074	1074	NUM
cana-5767	36	115	-	-	PUNCT
cana-5767	36	116	133x	133x	NUM
cana-5767	36	117	vol	vol	NOUN
cana-5767	36	118	32	32	NUM
cana-5767	36	119	no	no	NOUN
cana-5767	36	120	.	.	NOUN
cana-5767	36	121	1	1	NUM
cana-5767	36	122	(	(	PUNCT
cana-5767	36	123	2025	2025	NUM
cana-5767	36	124	)	)	PUNCT
cana-5767	36	125	562	562	NUM
cana-5767	36	126	https://internationalpubls.com	https://internationalpubls.com	X
cana-5767	36	127	#	#	NOUN
cana-5767	36	128	apply	apply	VERB
cana-5767	36	129	spectral	spectral	ADJ
cana-5767	36	130	clustering	clustering	NOUN
cana-5767	36	131	to	to	ADP
cana-5767	36	132	the	the	DET
cana-5767	36	133	perturbed	perturb	VERB
cana-5767	36	134	adjacency	adjacency	NOUN
cana-5767	36	135	matrix	matrix	NOUN
cana-5767	36	136	labels_perturbed	labels_perturbe	VERB
cana-5767	36	137	=	=	SYM
cana-5767	36	138	sc.fit_predict(adj_matrix_perturbed	sc.fit_predict(adj_matrix_perturbe	VERB
cana-5767	36	139	)	)	PUNCT
cana-5767	36	140	#	#	NOUN
cana-5767	36	141	calculate	calculate	NOUN
cana-5767	36	142	the	the	DET
cana-5767	36	143	adjusted	adjust	VERB
cana-5767	36	144	rand	rand	NOUN
cana-5767	36	145	index	index	NOUN
cana-5767	36	146	to	to	PART
cana-5767	36	147	compare	compare	VERB
cana-5767	36	148	the	the	DET
cana-5767	36	149	clustering	clustering	ADJ
cana-5767	36	150	results	result	NOUN
cana-5767	36	151	ari_score	ari_score	PUNCT
cana-5767	36	152	=	=	SYM
cana-5767	36	153	adjusted_rand_score(labels_original	adjusted_rand_score(labels_original	ADJ
cana-5767	36	154	,	,	PUNCT
cana-5767	36	155	labels_perturbed	labels_perturbe	VERB
cana-5767	36	156	)	)	PUNCT
cana-5767	36	157	print("adjusted	print("adjuste	VERB
cana-5767	36	158	rand	rand	NOUN
cana-5767	36	159	index	index	NOUN
cana-5767	36	160	score	score	NOUN
cana-5767	36	161	between	between	ADP
cana-5767	36	162	original	original	ADJ
cana-5767	36	163	and	and	CCONJ
cana-5767	36	164	perturbed	perturb	VERB
cana-5767	36	165	clustering	cluster	VERB
cana-5767	36	166	:	:	PUNCT
cana-5767	36	167	"	"	PUNCT
cana-5767	36	168	,	,	PUNCT
cana-5767	36	169	ari_score	ari_score	NOUN
cana-5767	36	170	)	)	PUNCT
cana-5767	36	171	#	#	NOUN
cana-5767	36	172	visualize	visualize	VERB
cana-5767	36	173	the	the	DET
cana-5767	36	174	original	original	ADJ
cana-5767	36	175	and	and	CCONJ
cana-5767	36	176	perturbed	perturb	VERB
cana-5767	36	177	graphs	graph	NOUN
cana-5767	36	178	plt.figure(figsize=(12	plt.figure(figsize=(12	NOUN
cana-5767	36	179	,	,	PUNCT
cana-5767	36	180	6	6	NUM
cana-5767	36	181	)	)	PUNCT
cana-5767	36	182	)	)	PUNCT
cana-5767	36	183	plt.subplot(1	plt.subplot(1	NOUN
cana-5767	36	184	,	,	PUNCT
cana-5767	36	185	2	2	NUM
cana-5767	36	186	,	,	PUNCT
cana-5767	36	187	1	1	NUM
cana-5767	36	188	)	)	PUNCT
cana-5767	36	189	nx.draw(graph	nx.draw(graph	PROPN
cana-5767	36	190	,	,	PUNCT
cana-5767	36	191	node_color	node_color	NOUN
cana-5767	36	192	=	=	SYM
cana-5767	36	193	labels_original	labels_original	ADJ
cana-5767	36	194	,	,	PUNCT
cana-5767	36	195	with_labels	with_label	NOUN
cana-5767	36	196	=	=	SYM
cana-5767	36	197	true	true	ADJ
cana-5767	36	198	,	,	PUNCT
cana-5767	36	199	cmap	cmap	NOUN
cana-5767	36	200	=	=	SYM
cana-5767	36	201	plt.cm.rainbow	plt.cm.rainbow	NOUN
cana-5767	36	202	)	)	PUNCT
cana-5767	36	203	plt.title("original	plt.title("original	ADJ
cana-5767	36	204	graph	graph	NOUN
cana-5767	36	205	with	with	ADP
cana-5767	36	206	spectral	spectral	ADJ
cana-5767	36	207	clustering	clustering	NOUN
cana-5767	36	208	"	"	PUNCT
cana-5767	36	209	)	)	PUNCT
cana-5767	36	210	plt.subplot(1	plt.subplot(1	NOUN
cana-5767	36	211	,	,	PUNCT
cana-5767	36	212	2	2	NUM
cana-5767	36	213	,	,	PUNCT
cana-5767	36	214	2	2	NUM
cana-5767	36	215	)	)	PUNCT
cana-5767	36	216	nx.draw(graph	nx.draw(graph	PROPN
cana-5767	36	217	,	,	PUNCT
cana-5767	36	218	node_color	node_color	NOUN
cana-5767	36	219	=	=	SYM
cana-5767	36	220	labels_perturbed	labels_perturbe	VERB
cana-5767	36	221	,	,	PUNCT
cana-5767	36	222	with_labels	with_label	NOUN
cana-5767	36	223	=	=	SYM
cana-5767	36	224	true	true	ADJ
cana-5767	36	225	,	,	PUNCT
cana-5767	36	226	cmap	cmap	NOUN
cana-5767	36	227	=	=	SYM
cana-5767	36	228	plt.cm.rainbow	plt.cm.rainbow	NOUN
cana-5767	36	229	)	)	PUNCT
cana-5767	36	230	plt.title("perturbed	plt.title("perturbe	VERB
cana-5767	36	231	graph	graph	NOUN
cana-5767	36	232	with	with	ADP
cana-5767	36	233	spectral	spectral	ADJ
cana-5767	36	234	clustering	clustering	NOUN
cana-5767	36	235	"	"	PUNCT
cana-5767	36	236	)	)	PUNCT
cana-5767	36	237	plt.show	plt.show	PRON
cana-5767	36	238	(	(	PUNCT
cana-5767	36	239	)	)	PUNCT
cana-5767	37	1	1	1	X
cana-5767	37	2	.	.	X
cana-5767	38	1	purpose	purpose	NOUN
cana-5767	38	2	of	of	ADP
cana-5767	38	3	the	the	DET
cana-5767	38	4	program	program	NOUN
cana-5767	38	5	this	this	DET
cana-5767	38	6	program	program	NOUN
cana-5767	38	7	explores	explore	VERB
cana-5767	38	8	spectral	spectral	ADJ
cana-5767	38	9	clustering	clustering	NOUN
cana-5767	38	10	(	(	PUNCT
cana-5767	38	11	a	a	DET
cana-5767	38	12	machine	machine	NOUN
cana-5767	38	13	learning	learning	NOUN
cana-5767	38	14	method	method	NOUN
cana-5767	38	15	)	)	PUNCT
cana-5767	38	16	applied	apply	VERB
cana-5767	38	17	to	to	ADP
cana-5767	38	18	a	a	DET
cana-5767	38	19	graph	graph	NOUN
cana-5767	38	20	.	.	PUNCT
cana-5767	39	1	it	it	PRON
cana-5767	39	2	examines	examine	VERB
cana-5767	39	3	how	how	SCONJ
cana-5767	39	4	perturbations	perturbation	NOUN
cana-5767	39	5	(	(	PUNCT
cana-5767	39	6	small	small	ADJ
cana-5767	39	7	noise	noise	NOUN
cana-5767	39	8	added	add	VERB
cana-5767	39	9	to	to	ADP
cana-5767	39	10	the	the	DET
cana-5767	39	11	graph	graph	NOUN
cana-5767	39	12	's	's	PART
cana-5767	39	13	adjacency	adjacency	NOUN
cana-5767	39	14	matrix	matrix	NOUN
cana-5767	39	15	)	)	PUNCT
cana-5767	39	16	affect	affect	VERB
cana-5767	39	17	the	the	DET
cana-5767	39	18	results	result	NOUN
cana-5767	39	19	of	of	ADP
cana-5767	39	20	spectral	spectral	ADJ
cana-5767	39	21	clustering	clustering	NOUN
cana-5767	39	22	.	.	PUNCT
cana-5767	40	1	2	2	X
cana-5767	40	2	.	.	X
cana-5767	40	3	key	key	ADJ
cana-5767	40	4	steps	step	NOUN
cana-5767	40	5	in	in	ADP
cana-5767	40	6	the	the	DET
cana-5767	40	7	program	program	NOUN
cana-5767	40	8	1	1	NUM
cana-5767	40	9	.	.	PUNCT
cana-5767	41	1	graph	graph	NOUN
cana-5767	41	2	creation	creation	NOUN
cana-5767	41	3	:	:	PUNCT
cana-5767	42	1	o	o	X
cana-5767	42	2	a	a	DET
cana-5767	42	3	random	random	ADJ
cana-5767	42	4	graph	graph	NOUN
cana-5767	42	5	with	with	ADP
cana-5767	42	6	100	100	NUM
cana-5767	42	7	nodes	node	NOUN
cana-5767	42	8	is	be	AUX
cana-5767	42	9	created	create	VERB
cana-5767	42	10	using	use	VERB
cana-5767	42	11	the	the	DET
cana-5767	42	12	erdős	erdős	NOUN
cana-5767	42	13	–	–	PUNCT
cana-5767	42	14	rényi	rényi	NOUN
cana-5767	42	15	model	model	NOUN
cana-5767	42	16	.	.	PUNCT
cana-5767	43	1	nodes	node	NOUN
cana-5767	43	2	represent	represent	VERB
cana-5767	43	3	data	datum	NOUN
cana-5767	43	4	points	point	NOUN
cana-5767	43	5	,	,	PUNCT
cana-5767	43	6	and	and	CCONJ
cana-5767	43	7	edges	edge	NOUN
cana-5767	43	8	indicate	indicate	VERB
cana-5767	43	9	connections	connection	NOUN
cana-5767	43	10	between	between	ADP
cana-5767	43	11	nodes	node	NOUN
cana-5767	43	12	.	.	PUNCT
cana-5767	44	1	communications	communication	NOUN
cana-5767	44	2	on	on	ADP
cana-5767	44	3	applied	apply	VERB
cana-5767	44	4	nonlinear	nonlinear	ADJ
cana-5767	44	5	analysis	analysis	NOUN
cana-5767	44	6	issn	issn	NOUN
cana-5767	44	7	:	:	PUNCT
cana-5767	44	8	1074	1074	NUM
cana-5767	44	9	-	-	PUNCT
cana-5767	44	10	133x	133x	NUM
cana-5767	44	11	vol	vol	NOUN
cana-5767	44	12	32	32	NUM
cana-5767	44	13	no	no	NOUN
cana-5767	44	14	.	.	NOUN
cana-5767	44	15	1	1	NUM
cana-5767	44	16	(	(	PUNCT
cana-5767	44	17	2025	2025	NUM
cana-5767	44	18	)	)	PUNCT
cana-5767	44	19	563	563	NUM
cana-5767	45	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5767	45	2	2	2	X
cana-5767	45	3	.	.	X
cana-5767	45	4	spectral	spectral	ADJ
cana-5767	45	5	clustering	clustering	NOUN
cana-5767	45	6	on	on	ADP
cana-5767	45	7	the	the	DET
cana-5767	45	8	original	original	ADJ
cana-5767	45	9	graph	graph	NOUN
cana-5767	45	10	:	:	PUNCT
cana-5767	45	11	o	o	NOUN
cana-5767	45	12	spectral	spectral	ADJ
cana-5767	45	13	clustering	cluster	VERB
cana-5767	45	14	groups	group	NOUN
cana-5767	45	15	nodes	node	NOUN
cana-5767	45	16	into	into	ADP
cana-5767	45	17	3	3	NUM
cana-5767	45	18	clusters	cluster	NOUN
cana-5767	45	19	based	base	VERB
cana-5767	45	20	on	on	ADP
cana-5767	45	21	the	the	DET
cana-5767	45	22	graph	graph	NOUN
cana-5767	45	23	's	's	PART
cana-5767	45	24	adjacency	adjacency	NOUN
cana-5767	45	25	matrix	matrix	NOUN
cana-5767	45	26	(	(	PUNCT
cana-5767	45	27	relationships	relationship	NOUN
cana-5767	45	28	between	between	ADP
cana-5767	45	29	nodes	node	NOUN
cana-5767	45	30	)	)	PUNCT
cana-5767	45	31	.	.	PUNCT
cana-5767	46	1	3	3	X
cana-5767	46	2	.	.	X
cana-5767	46	3	adding	add	VERB
cana-5767	46	4	perturbation	perturbation	NOUN
cana-5767	46	5	:	:	PUNCT
cana-5767	46	6	o	o	NOUN
cana-5767	46	7	small	small	ADJ
cana-5767	46	8	random	random	ADJ
cana-5767	46	9	gaussian	gaussian	NOUN
cana-5767	46	10	noise	noise	NOUN
cana-5767	46	11	is	be	AUX
cana-5767	46	12	added	add	VERB
cana-5767	46	13	to	to	ADP
cana-5767	46	14	the	the	DET
cana-5767	46	15	adjacency	adjacency	NOUN
cana-5767	46	16	matrix	matrix	NOUN
cana-5767	46	17	to	to	PART
cana-5767	46	18	simulate	simulate	VERB
cana-5767	46	19	a	a	DET
cana-5767	46	20	real	real	ADJ
cana-5767	46	21	-	-	PUNCT
cana-5767	46	22	world	world	NOUN
cana-5767	46	23	scenario	scenario	NOUN
cana-5767	46	24	where	where	SCONJ
cana-5767	46	25	data	datum	NOUN
cana-5767	46	26	might	might	AUX
cana-5767	46	27	be	be	AUX
cana-5767	46	28	noisy	noisy	ADJ
cana-5767	46	29	or	or	CCONJ
cana-5767	46	30	slightly	slightly	ADV
cana-5767	46	31	altered	alter	VERB
cana-5767	46	32	.	.	PUNCT
cana-5767	47	1	4	4	X
cana-5767	47	2	.	.	X
cana-5767	47	3	spectral	spectral	ADJ
cana-5767	47	4	clustering	clustering	NOUN
cana-5767	47	5	on	on	ADP
cana-5767	47	6	the	the	DET
cana-5767	47	7	perturbed	perturb	VERB
cana-5767	47	8	graph	graph	NOUN
cana-5767	47	9	:	:	PUNCT
cana-5767	47	10	o	o	NOUN
cana-5767	47	11	the	the	DET
cana-5767	47	12	clustering	clustering	NOUN
cana-5767	47	13	is	be	AUX
cana-5767	47	14	performed	perform	VERB
cana-5767	47	15	again	again	ADV
cana-5767	47	16	on	on	ADP
cana-5767	47	17	the	the	DET
cana-5767	47	18	noisy	noisy	ADJ
cana-5767	47	19	graph	graph	NOUN
cana-5767	47	20	to	to	PART
cana-5767	47	21	see	see	VERB
cana-5767	47	22	if	if	SCONJ
cana-5767	47	23	the	the	DET
cana-5767	47	24	results	result	NOUN
cana-5767	47	25	(	(	PUNCT
cana-5767	47	26	cluster	cluster	NOUN
cana-5767	47	27	assignments	assignment	NOUN
cana-5767	47	28	)	)	PUNCT
cana-5767	47	29	change	change	NOUN
cana-5767	47	30	.	.	PUNCT
cana-5767	48	1	5	5	X
cana-5767	48	2	.	.	NUM
cana-5767	48	3	adjusted	adjust	VERB
cana-5767	48	4	rand	rand	NOUN
cana-5767	48	5	index	index	NOUN
cana-5767	48	6	(	(	PUNCT
cana-5767	48	7	ari	ari	PROPN
cana-5767	48	8	):	):	PUNCT
cana-5767	48	9	o	o	NOUN
cana-5767	48	10	the	the	DET
cana-5767	48	11	ari	ari	PROPN
cana-5767	48	12	measures	measure	VERB
cana-5767	48	13	how	how	SCONJ
cana-5767	48	14	similar	similar	ADJ
cana-5767	48	15	the	the	DET
cana-5767	48	16	cluster	cluster	NOUN
cana-5767	48	17	assignments	assignment	NOUN
cana-5767	48	18	are	be	AUX
cana-5767	48	19	before	before	ADP
cana-5767	48	20	and	and	CCONJ
cana-5767	48	21	after	after	ADP
cana-5767	48	22	the	the	DET
cana-5767	48	23	perturbation	perturbation	NOUN
cana-5767	48	24	.	.	PUNCT
cana-5767	49	1	a	a	DET
cana-5767	49	2	higher	high	ADJ
cana-5767	49	3	ari	ari	PROPN
cana-5767	49	4	score	score	NOUN
cana-5767	49	5	(	(	PUNCT
cana-5767	49	6	closer	close	ADV
cana-5767	49	7	to	to	PART
cana-5767	49	8	1	1	NUM
cana-5767	49	9	)	)	PUNCT
cana-5767	49	10	means	mean	VERB
cana-5767	49	11	the	the	DET
cana-5767	49	12	clustering	clustering	NOUN
cana-5767	49	13	is	be	AUX
cana-5767	49	14	robust	robust	ADJ
cana-5767	49	15	to	to	ADP
cana-5767	49	16	perturbations	perturbation	NOUN
cana-5767	49	17	.	.	PUNCT
cana-5767	50	1	6	6	X
cana-5767	50	2	.	.	X
cana-5767	50	3	visualization	visualization	NOUN
cana-5767	50	4	:	:	PUNCT
cana-5767	50	5	o	o	NOUN
cana-5767	50	6	the	the	DET
cana-5767	50	7	program	program	NOUN
cana-5767	50	8	visualizes	visualize	VERB
cana-5767	50	9	the	the	DET
cana-5767	50	10	original	original	ADJ
cana-5767	50	11	and	and	CCONJ
cana-5767	50	12	perturbed	perturb	VERB
cana-5767	50	13	graph	graph	NOUN
cana-5767	50	14	clusters	cluster	NOUN
cana-5767	50	15	using	use	VERB
cana-5767	50	16	different	different	ADJ
cana-5767	50	17	colours	colour	NOUN
cana-5767	50	18	(	(	PUNCT
cana-5767	50	19	e.g.	e.g.	ADV
cana-5767	50	20	,	,	PUNCT
cana-5767	50	21	red	red	ADJ
cana-5767	50	22	,	,	PUNCT
cana-5767	50	23	green	green	ADJ
cana-5767	50	24	,	,	PUNCT
cana-5767	50	25	purple	purple	ADJ
cana-5767	50	26	)	)	PUNCT
cana-5767	50	27	.	.	PUNCT
cana-5767	51	1	3	3	X
cana-5767	51	2	.	.	X
cana-5767	51	3	key	key	ADJ
cana-5767	51	4	outputs	output	NOUN
cana-5767	51	5	●	●	PUNCT
cana-5767	51	6	ari	ari	PROPN
cana-5767	51	7	score	score	NOUN
cana-5767	51	8	:	:	PUNCT
cana-5767	51	9	o	o	NOUN
cana-5767	51	10	indicates	indicate	VERB
cana-5767	51	11	how	how	SCONJ
cana-5767	51	12	consistent	consistent	ADJ
cana-5767	51	13	the	the	DET
cana-5767	51	14	cluster	cluster	NOUN
cana-5767	51	15	assignments	assignment	NOUN
cana-5767	51	16	are	be	AUX
cana-5767	51	17	between	between	ADP
cana-5767	51	18	the	the	DET
cana-5767	51	19	original	original	ADJ
cana-5767	51	20	and	and	CCONJ
cana-5767	51	21	perturbed	perturb	VERB
cana-5767	51	22	graphs	graph	NOUN
cana-5767	51	23	.	.	PUNCT
cana-5767	52	1	for	for	ADP
cana-5767	52	2	example	example	NOUN
cana-5767	52	3	,	,	PUNCT
cana-5767	52	4	if	if	SCONJ
cana-5767	52	5	the	the	DET
cana-5767	52	6	ari	ari	PROPN
cana-5767	52	7	score	score	NOUN
cana-5767	52	8	is	be	AUX
cana-5767	52	9	0.9	0.9	NUM
cana-5767	52	10	,	,	PUNCT
cana-5767	52	11	the	the	DET
cana-5767	52	12	clustering	clustering	NOUN
cana-5767	52	13	is	be	AUX
cana-5767	52	14	robust	robust	ADJ
cana-5767	52	15	and	and	CCONJ
cana-5767	52	16	stable	stable	ADJ
cana-5767	52	17	under	under	ADP
cana-5767	52	18	small	small	ADJ
cana-5767	52	19	perturbations	perturbation	NOUN
cana-5767	52	20	.	.	PUNCT
cana-5767	53	1	●	●	PUNCT
cana-5767	53	2	graph	graph	NOUN
cana-5767	53	3	visualization	visualization	NOUN
cana-5767	53	4	:	:	PUNCT
cana-5767	53	5	o	o	NOUN
cana-5767	53	6	original	original	ADJ
cana-5767	53	7	graph	graph	NOUN
cana-5767	53	8	:	:	PUNCT
cana-5767	53	9	nodes	node	NOUN
cana-5767	53	10	are	be	AUX
cana-5767	53	11	colored	color	VERB
cana-5767	53	12	based	base	VERB
cana-5767	53	13	on	on	ADP
cana-5767	53	14	their	their	PRON
cana-5767	53	15	clusters	cluster	NOUN
cana-5767	53	16	in	in	ADP
cana-5767	53	17	the	the	DET
cana-5767	53	18	unperturbed	unperturbed	ADJ
cana-5767	53	19	graph	graph	NOUN
cana-5767	53	20	.	.	PUNCT
cana-5767	54	1	o	o	NOUN
cana-5767	54	2	perturbed	perturb	VERB
cana-5767	54	3	graph	graph	NOUN
cana-5767	54	4	:	:	PUNCT
cana-5767	54	5	nodes	node	NOUN
cana-5767	54	6	are	be	AUX
cana-5767	54	7	colored	color	VERB
cana-5767	54	8	based	base	VERB
cana-5767	54	9	on	on	ADP
cana-5767	54	10	their	their	PRON
cana-5767	54	11	clusters	cluster	NOUN
cana-5767	54	12	in	in	ADP
cana-5767	54	13	the	the	DET
cana-5767	54	14	noisy	noisy	ADJ
cana-5767	54	15	graph	graph	NOUN
cana-5767	54	16	.	.	PUNCT
cana-5767	55	1	4	4	X
cana-5767	55	2	.	.	X
cana-5767	55	3	role	role	NOUN
cana-5767	55	4	of	of	ADP
cana-5767	55	5	colours	colour	NOUN
cana-5767	55	6	in	in	ADP
cana-5767	55	7	graphs	graph	NOUN
cana-5767	55	8	the	the	DET
cana-5767	55	9	colours	colour	NOUN
cana-5767	55	10	(	(	PUNCT
cana-5767	55	11	red	red	ADJ
cana-5767	55	12	,	,	PUNCT
cana-5767	55	13	green	green	ADJ
cana-5767	55	14	,	,	PUNCT
cana-5767	55	15	purple	purple	ADJ
cana-5767	55	16	,	,	PUNCT
cana-5767	55	17	etc	etc	X
cana-5767	55	18	.	.	X
cana-5767	55	19	)	)	PUNCT
cana-5767	55	20	represent	represent	VERB
cana-5767	55	21	the	the	DET
cana-5767	55	22	clusters	cluster	NOUN
cana-5767	55	23	identified	identify	VERB
cana-5767	55	24	by	by	ADP
cana-5767	55	25	the	the	DET
cana-5767	55	26	spectral	spectral	ADJ
cana-5767	55	27	clustering	clustering	ADJ
cana-5767	55	28	algorithm	algorithm	NOUN
cana-5767	55	29	.	.	PUNCT
cana-5767	56	1	nodes	node	NOUN
cana-5767	56	2	with	with	ADP
cana-5767	56	3	the	the	DET
cana-5767	56	4	same	same	ADJ
cana-5767	56	5	colour	colour	NOUN
cana-5767	56	6	belong	belong	VERB
cana-5767	56	7	to	to	ADP
cana-5767	56	8	the	the	DET
cana-5767	56	9	same	same	ADJ
cana-5767	56	10	cluster	cluster	NOUN
cana-5767	56	11	.	.	PUNCT
cana-5767	57	1	differences	difference	NOUN
cana-5767	57	2	in	in	ADP
cana-5767	57	3	colours	colour	NOUN
cana-5767	57	4	between	between	ADP
cana-5767	57	5	the	the	DET
cana-5767	57	6	two	two	NUM
cana-5767	57	7	graphs	graph	NOUN
cana-5767	57	8	(	(	PUNCT
cana-5767	57	9	original	original	ADJ
cana-5767	57	10	and	and	CCONJ
cana-5767	57	11	perturbed	perturb	VERB
cana-5767	57	12	)	)	PUNCT
cana-5767	57	13	indicate	indicate	VERB
cana-5767	57	14	changes	change	NOUN
cana-5767	57	15	in	in	ADP
cana-5767	57	16	clustering	clustering	ADJ
cana-5767	57	17	results	result	NOUN
cana-5767	57	18	due	due	ADP
cana-5767	57	19	to	to	ADP
cana-5767	57	20	noise	noise	NOUN
cana-5767	57	21	.	.	PUNCT
cana-5767	58	1	5	5	X
cana-5767	58	2	.	.	X
cana-5767	58	3	research	research	NOUN
cana-5767	58	4	significance	significance	NOUN
cana-5767	58	5	this	this	DET
cana-5767	58	6	program	program	NOUN
cana-5767	58	7	connects	connect	VERB
cana-5767	58	8	to	to	ADP
cana-5767	58	9	the	the	DET
cana-5767	58	10	research	research	NOUN
cana-5767	58	11	by	by	ADP
cana-5767	58	12	:	:	PUNCT
cana-5767	58	13	communications	communication	NOUN
cana-5767	58	14	on	on	ADP
cana-5767	58	15	applied	apply	VERB
cana-5767	58	16	nonlinear	nonlinear	ADJ
cana-5767	58	17	analysis	analysis	NOUN
cana-5767	58	18	issn	issn	NOUN
cana-5767	58	19	:	:	PUNCT
cana-5767	58	20	1074	1074	NUM
cana-5767	58	21	-	-	PUNCT
cana-5767	58	22	133x	133x	NUM
cana-5767	58	23	vol	vol	NOUN
cana-5767	58	24	32	32	NUM
cana-5767	58	25	no	no	NOUN
cana-5767	58	26	.	.	NOUN
cana-5767	58	27	1	1	NUM
cana-5767	58	28	(	(	PUNCT
cana-5767	58	29	2025	2025	NUM
cana-5767	58	30	)	)	PUNCT
cana-5767	59	1	564	564	NUM
cana-5767	59	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-5767	59	3	●	●	PUNCT
cana-5767	59	4	validating	validate	VERB
cana-5767	59	5	robustness	robustness	NOUN
cana-5767	59	6	:	:	PUNCT
cana-5767	59	7	by	by	ADP
cana-5767	59	8	showing	show	VERB
cana-5767	59	9	that	that	SCONJ
cana-5767	59	10	the	the	DET
cana-5767	59	11	clustering	clustering	NOUN
cana-5767	59	12	remains	remain	VERB
cana-5767	59	13	consistent	consistent	ADJ
cana-5767	59	14	under	under	ADP
cana-5767	59	15	perturbations	perturbation	NOUN
cana-5767	59	16	(	(	PUNCT
cana-5767	59	17	high	high	ADJ
cana-5767	59	18	ari	ari	PROPN
cana-5767	59	19	score	score	NOUN
cana-5767	59	20	)	)	PUNCT
cana-5767	59	21	,	,	PUNCT
cana-5767	59	22	it	it	PRON
cana-5767	59	23	demonstrates	demonstrate	VERB
cana-5767	59	24	the	the	DET
cana-5767	59	25	stability	stability	NOUN
cana-5767	59	26	of	of	ADP
cana-5767	59	27	spectral	spectral	ADJ
cana-5767	59	28	methods	method	NOUN
cana-5767	59	29	.	.	PUNCT
cana-5767	60	1	●	●	PUNCT
cana-5767	60	2	using	use	VERB
cana-5767	60	3	operator	operator	NOUN
cana-5767	60	4	theory	theory	NOUN
cana-5767	60	5	:	:	PUNCT
cana-5767	60	6	the	the	DET
cana-5767	60	7	stability	stability	NOUN
cana-5767	60	8	of	of	ADP
cana-5767	60	9	eigenvalues	eigenvalue	NOUN
cana-5767	60	10	of	of	ADP
cana-5767	60	11	the	the	DET
cana-5767	60	12	graph	graph	NOUN
cana-5767	60	13	laplacian	laplacian	NOUN
cana-5767	60	14	(	(	PUNCT
cana-5767	60	15	the	the	DET
cana-5767	60	16	operator	operator	NOUN
cana-5767	60	17	)	)	PUNCT
cana-5767	60	18	under	under	ADP
cana-5767	60	19	small	small	ADJ
cana-5767	60	20	perturbations	perturbation	NOUN
cana-5767	60	21	explains	explain	VERB
cana-5767	60	22	why	why	SCONJ
cana-5767	60	23	clustering	clustering	NOUN
cana-5767	60	24	is	be	AUX
cana-5767	60	25	robust	robust	ADJ
cana-5767	60	26	,	,	PUNCT
cana-5767	60	27	as	as	SCONJ
cana-5767	60	28	predicted	predict	VERB
cana-5767	60	29	by	by	ADP
cana-5767	60	30	weyl	weyl	PROPN
cana-5767	60	31	's	's	PART
cana-5767	60	32	theorem	theorem	ADJ
cana-5767	60	33	.	.	PROPN
cana-5767	60	34	1	1	X
cana-5767	60	35	.	.	X
cana-5767	60	36	stability	stability	NOUN
cana-5767	60	37	of	of	ADP
cana-5767	60	38	eigenvalues	eigenvalues	PROPN
cana-5767	60	39	(	(	PUNCT
cana-5767	60	40	weyl	weyl	PROPN
cana-5767	60	41	's	's	PART
cana-5767	60	42	theorem	theorem	ADJ
cana-5767	60	43	)	)	PUNCT
cana-5767	60	44	●	●	PRON
cana-5767	60	45	theorem	theorem	ADJ
cana-5767	60	46	link	link	NOUN
cana-5767	60	47	:	:	PUNCT
cana-5767	60	48	weyl	weyl	PROPN
cana-5767	60	49	's	's	PART
cana-5767	60	50	theorem	theorem	NOUN
cana-5767	60	51	ensures	ensure	VERB
cana-5767	60	52	that	that	SCONJ
cana-5767	60	53	the	the	DET
cana-5767	60	54	eigenvalues	eigenvalue	NOUN
cana-5767	60	55	of	of	ADP
cana-5767	60	56	the	the	DET
cana-5767	60	57	graph	graph	NOUN
cana-5767	60	58	laplacian	laplacian	NOUN
cana-5767	60	59	(	(	PUNCT
cana-5767	60	60	used	use	VERB
cana-5767	60	61	in	in	ADP
cana-5767	60	62	spectral	spectral	ADJ
cana-5767	60	63	clustering	clustering	NOUN
cana-5767	60	64	)	)	PUNCT
cana-5767	60	65	are	be	AUX
cana-5767	60	66	stable	stable	ADJ
cana-5767	60	67	under	under	ADP
cana-5767	60	68	small	small	ADJ
cana-5767	60	69	perturbations	perturbation	NOUN
cana-5767	60	70	.	.	PUNCT
cana-5767	61	1	in	in	ADP
cana-5767	61	2	your	your	PRON
cana-5767	61	3	program	program	NOUN
cana-5767	61	4	:	:	PUNCT
cana-5767	61	5	o	o	X
cana-5767	61	6	the	the	DET
cana-5767	61	7	adjacency	adjacency	NOUN
cana-5767	61	8	matrix	matrix	NOUN
cana-5767	61	9	is	be	AUX
cana-5767	61	10	perturbed	perturb	VERB
cana-5767	61	11	by	by	ADP
cana-5767	61	12	adding	add	VERB
cana-5767	61	13	random	random	ADJ
cana-5767	61	14	gaussian	gaussian	NOUN
cana-5767	61	15	noise	noise	NOUN
cana-5767	61	16	.	.	PUNCT
cana-5767	62	1	o	o	X
cana-5767	63	1	the	the	DET
cana-5767	63	2	theorem	theorem	NOUN
cana-5767	63	3	guarantees	guarantee	VERB
cana-5767	63	4	that	that	SCONJ
cana-5767	63	5	the	the	DET
cana-5767	63	6	eigenvalues	eigenvalue	NOUN
cana-5767	63	7	of	of	ADP
cana-5767	63	8	the	the	DET
cana-5767	63	9	laplacian	laplacian	ADJ
cana-5767	63	10	derived	derive	VERB
cana-5767	63	11	from	from	ADP
cana-5767	63	12	the	the	DET
cana-5767	63	13	perturbed	perturb	VERB
cana-5767	63	14	matrix	matrix	NOUN
cana-5767	63	15	remain	remain	VERB
cana-5767	63	16	close	close	ADJ
cana-5767	63	17	to	to	ADP
cana-5767	63	18	those	those	PRON
cana-5767	63	19	of	of	ADP
cana-5767	63	20	the	the	DET
cana-5767	63	21	original	original	ADJ
cana-5767	63	22	laplacian	laplacian	NOUN
cana-5767	63	23	,	,	PUNCT
cana-5767	63	24	provided	provide	VERB
cana-5767	63	25	the	the	DET
cana-5767	63	26	noise	noise	NOUN
cana-5767	63	27	magnitude	magnitude	NOUN
cana-5767	63	28	is	be	AUX
cana-5767	63	29	small	small	ADJ
cana-5767	63	30	.	.	PUNCT
cana-5767	64	1	●	●	PUNCT
cana-5767	64	2	program	program	NOUN
cana-5767	64	3	outcome	outcome	NOUN
cana-5767	64	4	:	:	PUNCT
cana-5767	64	5	because	because	SCONJ
cana-5767	64	6	eigenvalues	eigenvalue	NOUN
cana-5767	64	7	are	be	AUX
cana-5767	64	8	stable	stable	ADJ
cana-5767	64	9	,	,	PUNCT
cana-5767	64	10	the	the	DET
cana-5767	64	11	clustering	clustering	ADJ
cana-5767	64	12	structure	structure	NOUN
cana-5767	64	13	derived	derive	VERB
cana-5767	64	14	from	from	ADP
cana-5767	64	15	the	the	DET
cana-5767	64	16	eigenvectors	eigenvector	NOUN
cana-5767	64	17	of	of	ADP
cana-5767	64	18	the	the	DET
cana-5767	64	19	laplacian	laplacian	NOUN
cana-5767	64	20	remains	remain	VERB
cana-5767	64	21	consistent	consistent	ADJ
cana-5767	64	22	.	.	PUNCT
cana-5767	65	1	this	this	PRON
cana-5767	65	2	is	be	AUX
cana-5767	65	3	observed	observe	VERB
cana-5767	65	4	through	through	ADP
cana-5767	65	5	the	the	DET
cana-5767	65	6	high	high	ADJ
cana-5767	65	7	adjusted	adjust	VERB
cana-5767	65	8	rand	rand	NOUN
cana-5767	65	9	index	index	NOUN
cana-5767	65	10	(	(	PUNCT
cana-5767	65	11	ari	ari	PROPN
cana-5767	65	12	)	)	PUNCT
cana-5767	65	13	score	score	NOUN
cana-5767	65	14	in	in	ADP
cana-5767	65	15	your	your	PRON
cana-5767	65	16	program	program	NOUN
cana-5767	65	17	,	,	PUNCT
cana-5767	65	18	indicating	indicate	VERB
cana-5767	65	19	that	that	SCONJ
cana-5767	65	20	the	the	DET
cana-5767	65	21	cluster	cluster	NOUN
cana-5767	65	22	assignments	assignment	NOUN
cana-5767	65	23	do	do	AUX
cana-5767	65	24	not	not	PART
cana-5767	65	25	change	change	VERB
cana-5767	65	26	significantly	significantly	ADV
cana-5767	65	27	after	after	ADP
cana-5767	65	28	perturbation	perturbation	NOUN
cana-5767	65	29	.	.	PUNCT
cana-5767	65	30	.	.	PUNCT
cana-5767	66	1	stability	stability	NOUN
cana-5767	66	2	of	of	ADP
cana-5767	66	3	eigenspaces	eigenspace	NOUN
cana-5767	66	4	(	(	PUNCT
cana-5767	66	5	davis	davis	PROPN
cana-5767	66	6	-	-	PUNCT
cana-5767	66	7	kahan	kahan	NOUN
cana-5767	66	8	theorem	theorem	ADJ
cana-5767	66	9	)	)	PUNCT
cana-5767	66	10	●	●	PRON
cana-5767	66	11	theorem	theorem	ADJ
cana-5767	66	12	link	link	NOUN
cana-5767	66	13	:	:	PUNCT
cana-5767	66	14	the	the	DET
cana-5767	66	15	davis	davis	PROPN
cana-5767	66	16	-	-	PUNCT
cana-5767	66	17	kahan	kahan	PROPN
cana-5767	66	18	theorem	theorem	NOUN
cana-5767	66	19	bounds	bound	VERB
cana-5767	66	20	the	the	DET
cana-5767	66	21	difference	difference	NOUN
cana-5767	66	22	between	between	ADP
cana-5767	66	23	the	the	DET
cana-5767	66	24	eigenspaces	eigenspace	NOUN
cana-5767	66	25	of	of	ADP
cana-5767	66	26	the	the	DET
cana-5767	66	27	original	original	ADJ
cana-5767	66	28	and	and	CCONJ
cana-5767	66	29	perturbed	perturb	VERB
cana-5767	66	30	laplacians	laplacian	NOUN
cana-5767	66	31	.	.	PUNCT
cana-5767	67	1	the	the	DET
cana-5767	67	2	difference	difference	NOUN
cana-5767	67	3	depends	depend	VERB
cana-5767	67	4	on	on	ADP
cana-5767	67	5	the	the	DET
cana-5767	67	6	size	size	NOUN
cana-5767	67	7	of	of	ADP
cana-5767	67	8	the	the	DET
cana-5767	67	9	perturbation	perturbation	NOUN
cana-5767	67	10	and	and	CCONJ
cana-5767	67	11	the	the	DET
cana-5767	67	12	spectral	spectral	ADJ
cana-5767	67	13	gap	gap	NOUN
cana-5767	67	14	(	(	PUNCT
cana-5767	67	15	difference	difference	NOUN
cana-5767	67	16	between	between	ADP
cana-5767	67	17	consecutive	consecutive	ADJ
cana-5767	67	18	eigenvalues	eigenvalue	NOUN
cana-5767	67	19	)	)	PUNCT
cana-5767	67	20	.	.	PUNCT
cana-5767	68	1	in	in	ADP
cana-5767	68	2	your	your	PRON
cana-5767	68	3	program	program	NOUN
cana-5767	68	4	:	:	PUNCT
cana-5767	68	5	o	o	X
cana-5767	68	6	the	the	DET
cana-5767	68	7	spectral	spectral	ADJ
cana-5767	68	8	clustering	clustering	ADJ
cana-5767	68	9	algorithm	algorithm	NOUN
cana-5767	68	10	uses	use	VERB
cana-5767	68	11	the	the	DET
cana-5767	68	12	first	first	ADJ
cana-5767	68	13	k	k	PROPN
cana-5767	68	14	eigenvectors	eigenvector	NOUN
cana-5767	68	15	of	of	ADP
cana-5767	68	16	the	the	DET
cana-5767	68	17	laplacian	laplacian	NOUN
cana-5767	68	18	to	to	PART
cana-5767	68	19	embed	embed	VERB
cana-5767	68	20	nodes	node	NOUN
cana-5767	68	21	into	into	ADP
cana-5767	68	22	a	a	DET
cana-5767	68	23	low	low	ADJ
cana-5767	68	24	-	-	PUNCT
cana-5767	68	25	dimensional	dimensional	ADJ
cana-5767	68	26	space	space	NOUN
cana-5767	68	27	.	.	PUNCT
cana-5767	69	1	o	o	X
cana-5767	69	2	the	the	DET
cana-5767	69	3	theorem	theorem	NOUN
cana-5767	69	4	ensures	ensure	VERB
cana-5767	69	5	that	that	SCONJ
cana-5767	69	6	these	these	DET
cana-5767	69	7	embeddings	embedding	NOUN
cana-5767	69	8	remain	remain	VERB
cana-5767	69	9	close	close	ADJ
cana-5767	69	10	under	under	ADP
cana-5767	69	11	small	small	ADJ
cana-5767	69	12	perturbations	perturbation	NOUN
cana-5767	69	13	.	.	PUNCT
cana-5767	70	1	●	●	PUNCT
cana-5767	70	2	program	program	NOUN
cana-5767	70	3	outcome	outcome	NOUN
cana-5767	70	4	:	:	PUNCT
cana-5767	70	5	the	the	DET
cana-5767	70	6	clustering	clustering	ADJ
cana-5767	70	7	results	result	NOUN
cana-5767	70	8	remain	remain	VERB
cana-5767	70	9	robust	robust	ADJ
cana-5767	70	10	because	because	SCONJ
cana-5767	70	11	the	the	DET
cana-5767	70	12	perturbed	perturb	VERB
cana-5767	70	13	eigenvectors	eigenvector	NOUN
cana-5767	70	14	still	still	ADV
cana-5767	70	15	reflect	reflect	VERB
cana-5767	70	16	the	the	DET
cana-5767	70	17	original	original	ADJ
cana-5767	70	18	clustering	clustering	NOUN
cana-5767	70	19	structure	structure	NOUN
cana-5767	70	20	.	.	PUNCT
cana-5767	71	1	the	the	DET
cana-5767	71	2	high	high	ADJ
cana-5767	71	3	ari	ari	PROPN
cana-5767	71	4	score	score	NOUN
cana-5767	71	5	confirms	confirm	VERB
cana-5767	71	6	this	this	DET
cana-5767	71	7	stability	stability	NOUN
cana-5767	71	8	.	.	PUNCT
cana-5767	72	1	3	3	X
cana-5767	72	2	.	.	NUM
cana-5767	72	3	adjusted	adjust	VERB
cana-5767	72	4	rand	rand	NOUN
cana-5767	72	5	index	index	NOUN
cana-5767	72	6	(	(	PUNCT
cana-5767	72	7	ari	ari	PROPN
cana-5767	72	8	)	)	PUNCT
cana-5767	72	9	as	as	ADP
cana-5767	72	10	a	a	DET
cana-5767	72	11	practical	practical	ADJ
cana-5767	72	12	measure	measure	NOUN
cana-5767	72	13	●	●	X
cana-5767	72	14	theorem	theorem	ADJ
cana-5767	72	15	link	link	NOUN
cana-5767	72	16	:	:	PUNCT
cana-5767	72	17	the	the	DET
cana-5767	72	18	theoretical	theoretical	ADJ
cana-5767	72	19	bounds	bound	NOUN
cana-5767	72	20	from	from	ADP
cana-5767	72	21	weyl	weyl	PROPN
cana-5767	72	22	's	's	PART
cana-5767	72	23	and	and	CCONJ
cana-5767	72	24	davis	davis	PROPN
cana-5767	72	25	-	-	PUNCT
cana-5767	72	26	kahan	kahan	PROPN
cana-5767	72	27	's	's	PART
cana-5767	72	28	theorems	theorem	NOUN
cana-5767	72	29	ensure	ensure	VERB
cana-5767	72	30	that	that	SCONJ
cana-5767	72	31	the	the	DET
cana-5767	72	32	perturbations	perturbation	NOUN
cana-5767	72	33	cause	cause	VERB
cana-5767	72	34	minimal	minimal	ADJ
cana-5767	72	35	changes	change	NOUN
cana-5767	72	36	to	to	ADP
cana-5767	72	37	the	the	DET
cana-5767	72	38	eigenvalues	eigenvalue	NOUN
cana-5767	72	39	and	and	CCONJ
cana-5767	72	40	eigenspaces	eigenspace	NOUN
cana-5767	72	41	,	,	PUNCT
cana-5767	72	42	which	which	PRON
cana-5767	72	43	translates	translate	VERB
cana-5767	72	44	to	to	PART
cana-5767	72	45	minimal	minimal	ADJ
cana-5767	72	46	changes	change	NOUN
cana-5767	72	47	in	in	ADP
cana-5767	72	48	cluster	cluster	NOUN
cana-5767	72	49	assignments	assignment	NOUN
cana-5767	72	50	.	.	PUNCT
cana-5767	73	1	the	the	DET
cana-5767	73	2	ari	ari	PROPN
cana-5767	73	3	score	score	NOUN
cana-5767	73	4	quantifies	quantifie	NOUN
cana-5767	73	5	this	this	DET
cana-5767	73	6	similarity	similarity	NOUN
cana-5767	73	7	in	in	ADP
cana-5767	73	8	practice	practice	NOUN
cana-5767	73	9	.	.	PUNCT
cana-5767	74	1	●	●	PUNCT
cana-5767	74	2	program	program	NOUN
cana-5767	74	3	outcome	outcome	NOUN
cana-5767	74	4	:	:	PUNCT
cana-5767	74	5	your	your	PRON
cana-5767	74	6	program	program	NOUN
cana-5767	74	7	measures	measure	VERB
cana-5767	74	8	the	the	DET
cana-5767	74	9	ari	ari	PROPN
cana-5767	74	10	score	score	NOUN
cana-5767	74	11	to	to	PART
cana-5767	74	12	confirm	confirm	VERB
cana-5767	74	13	the	the	DET
cana-5767	74	14	robustness	robustness	NOUN
cana-5767	74	15	of	of	ADP
cana-5767	74	16	clustering	clustering	NOUN
cana-5767	74	17	.	.	PUNCT
cana-5767	75	1	a	a	DET
cana-5767	75	2	high	high	ADJ
cana-5767	75	3	ari	ari	PROPN
cana-5767	75	4	score	score	NOUN
cana-5767	75	5	provides	provide	VERB
cana-5767	75	6	empirical	empirical	ADJ
cana-5767	75	7	evidence	evidence	NOUN
cana-5767	75	8	for	for	ADP
cana-5767	75	9	the	the	DET
cana-5767	75	10	theoretical	theoretical	ADJ
cana-5767	75	11	stability	stability	NOUN
cana-5767	75	12	guaranteed	guarantee	VERB
cana-5767	75	13	by	by	ADP
cana-5767	75	14	the	the	DET
cana-5767	75	15	theorems	theorem	NOUN
cana-5767	75	16	.	.	PUNCT
cana-5767	76	1	communications	communication	NOUN
cana-5767	76	2	on	on	ADP
cana-5767	76	3	applied	apply	VERB
cana-5767	76	4	nonlinear	nonlinear	ADJ
cana-5767	76	5	analysis	analysis	NOUN
cana-5767	76	6	issn	issn	NOUN
cana-5767	76	7	:	:	PUNCT
cana-5767	76	8	1074	1074	NUM
cana-5767	76	9	-	-	PUNCT
cana-5767	76	10	133x	133x	NUM
cana-5767	76	11	vol	vol	NOUN
cana-5767	76	12	32	32	NUM
cana-5767	76	13	no	no	NOUN
cana-5767	76	14	.	.	NOUN
cana-5767	76	15	1	1	NUM
cana-5767	76	16	(	(	PUNCT
cana-5767	76	17	2025	2025	NUM
cana-5767	76	18	)	)	PUNCT
cana-5767	76	19	565	565	NUM
cana-5767	76	20	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-5767	76	21	4	4	X
cana-5767	76	22	.	.	PUNCT
cana-5767	76	23	spectral	spectral	ADJ
cana-5767	76	24	gap	gap	NOUN
cana-5767	76	25	and	and	CCONJ
cana-5767	76	26	robustness	robustness	NOUN
cana-5767	76	27	●	●	NUM
cana-5767	76	28	theorem	theorem	ADJ
cana-5767	76	29	link	link	NOUN
cana-5767	76	30	:	:	PUNCT
cana-5767	76	31	the	the	DET
cana-5767	76	32	robustness	robustness	NOUN
cana-5767	76	33	of	of	ADP
cana-5767	76	34	spectral	spectral	ADJ
cana-5767	76	35	clustering	clustering	NOUN
cana-5767	76	36	depends	depend	VERB
cana-5767	76	37	on	on	ADP
cana-5767	76	38	the	the	DET
cana-5767	76	39	size	size	NOUN
cana-5767	76	40	of	of	ADP
cana-5767	76	41	the	the	DET
cana-5767	76	42	spectral	spectral	ADJ
cana-5767	76	43	gap	gap	NOUN
cana-5767	76	44	,	,	PUNCT
cana-5767	76	45	which	which	PRON
cana-5767	76	46	is	be	AUX
cana-5767	76	47	the	the	DET
cana-5767	76	48	difference	difference	NOUN
cana-5767	76	49	between	between	ADP
cana-5767	76	50	the	the	DET
cana-5767	76	51	k	k	NOUN
cana-5767	76	52	-	-	PUNCT
cana-5767	76	53	th	th	X
cana-5767	76	54	and	and	CCONJ
cana-5767	76	55	(	(	PUNCT
cana-5767	76	56	k+1)-th	k+1)-th	NOUN
cana-5767	76	57	eigenvalues	eigenvalue	VERB
cana-5767	76	58	.	.	PUNCT
cana-5767	77	1	a	a	DET
cana-5767	77	2	larger	large	ADJ
cana-5767	77	3	spectral	spectral	ADJ
cana-5767	77	4	gap	gap	NOUN
cana-5767	77	5	improves	improve	VERB
cana-5767	77	6	robustness	robustness	NOUN
cana-5767	77	7	to	to	ADP
cana-5767	77	8	perturbations	perturbation	NOUN
cana-5767	77	9	.	.	PUNCT
cana-5767	78	1	●	●	PUNCT
cana-5767	78	2	program	program	NOUN
cana-5767	78	3	connection	connection	NOUN
cana-5767	78	4	:	:	PUNCT
cana-5767	78	5	in	in	ADP
cana-5767	78	6	your	your	PRON
cana-5767	78	7	program	program	NOUN
cana-5767	78	8	:	:	PUNCT
cana-5767	78	9	o	o	X
cana-5767	78	10	the	the	DET
cana-5767	78	11	spectral	spectral	ADJ
cana-5767	78	12	gap	gap	NOUN
cana-5767	78	13	determines	determine	VERB
cana-5767	78	14	the	the	DET
cana-5767	78	15	separation	separation	NOUN
cana-5767	78	16	between	between	ADP
cana-5767	78	17	clusters	cluster	NOUN
cana-5767	78	18	.	.	PUNCT
cana-5767	79	1	for	for	ADP
cana-5767	79	2	well	well	ADV
cana-5767	79	3	-	-	PUNCT
cana-5767	79	4	separated	separate	VERB
cana-5767	79	5	clusters	cluster	NOUN
cana-5767	79	6	,	,	PUNCT
cana-5767	79	7	the	the	DET
cana-5767	79	8	spectral	spectral	ADJ
cana-5767	79	9	gap	gap	NOUN
cana-5767	79	10	is	be	AUX
cana-5767	79	11	large	large	ADJ
cana-5767	79	12	,	,	PUNCT
cana-5767	79	13	leading	lead	VERB
cana-5767	79	14	to	to	ADP
cana-5767	79	15	more	more	ADV
cana-5767	79	16	robust	robust	ADJ
cana-5767	79	17	clustering	clustering	NOUN
cana-5767	79	18	.	.	PUNCT
cana-5767	80	1	o	o	INTJ
cana-5767	80	2	this	this	PRON
cana-5767	80	3	is	be	AUX
cana-5767	80	4	why	why	SCONJ
cana-5767	80	5	the	the	DET
cana-5767	80	6	clustering	clustering	NOUN
cana-5767	80	7	remains	remain	VERB
cana-5767	80	8	stable	stable	ADJ
cana-5767	80	9	despite	despite	SCONJ
cana-5767	80	10	the	the	DET
cana-5767	80	11	addition	addition	NOUN
cana-5767	80	12	of	of	ADP
cana-5767	80	13	gaussian	gaussian	ADJ
cana-5767	80	14	noise	noise	NOUN
cana-5767	80	15	.	.	PUNCT
cana-5767	81	1	summary	summary	NOUN
cana-5767	81	2	1	1	NUM
cana-5767	81	3	.	.	PUNCT
cana-5767	82	1	weyl	weyl	PROPN
cana-5767	82	2	's	's	PART
cana-5767	82	3	theorem	theorem	NOUN
cana-5767	82	4	explains	explain	VERB
cana-5767	82	5	the	the	DET
cana-5767	82	6	stability	stability	NOUN
cana-5767	82	7	of	of	ADP
cana-5767	82	8	eigenvalues	eigenvalue	NOUN
cana-5767	82	9	under	under	ADP
cana-5767	82	10	perturbations	perturbation	NOUN
cana-5767	82	11	in	in	ADP
cana-5767	82	12	the	the	DET
cana-5767	82	13	adjacency	adjacency	NOUN
cana-5767	82	14	matrix	matrix	NOUN
cana-5767	82	15	.	.	PUNCT
cana-5767	83	1	2	2	X
cana-5767	83	2	.	.	X
cana-5767	83	3	davis	davis	PROPN
cana-5767	83	4	-	-	PUNCT
cana-5767	83	5	kahan	kahan	PROPN
cana-5767	83	6	theorem	theorem	VERB
cana-5767	83	7	guarantees	guarantee	NOUN
cana-5767	83	8	that	that	SCONJ
cana-5767	83	9	the	the	DET
cana-5767	83	10	eigenvectors	eigenvector	NOUN
cana-5767	83	11	and	and	CCONJ
cana-5767	83	12	the	the	DET
cana-5767	83	13	clustering	clustering	NOUN
cana-5767	83	14	derived	derive	VERB
cana-5767	83	15	from	from	ADP
cana-5767	83	16	them	they	PRON
cana-5767	83	17	remain	remain	VERB
cana-5767	83	18	consistent	consistent	ADJ
cana-5767	83	19	under	under	ADP
cana-5767	83	20	small	small	ADJ
cana-5767	83	21	perturbations	perturbation	NOUN
cana-5767	83	22	.	.	PUNCT
cana-5767	84	1	3	3	X
cana-5767	84	2	.	.	X
cana-5767	84	3	your	your	PRON
cana-5767	84	4	program	program	NOUN
cana-5767	84	5	demonstrates	demonstrate	VERB
cana-5767	84	6	these	these	DET
cana-5767	84	7	theoretical	theoretical	ADJ
cana-5767	84	8	results	result	NOUN
cana-5767	84	9	by	by	ADP
cana-5767	84	10	showing	show	VERB
cana-5767	84	11	:	:	PUNCT
cana-5767	84	12	o	o	NOUN
cana-5767	84	13	a	a	DET
cana-5767	84	14	high	high	ADJ
cana-5767	84	15	ari	ari	PROPN
cana-5767	84	16	score	score	NOUN
cana-5767	84	17	.	.	PUNCT
cana-5767	85	1	o	o	X
cana-5767	85	2	minimal	minimal	ADJ
cana-5767	85	3	changes	change	NOUN
cana-5767	85	4	in	in	ADP
cana-5767	85	5	clustering	clustering	ADJ
cana-5767	85	6	assignments	assignment	NOUN
cana-5767	85	7	between	between	ADP
cana-5767	85	8	the	the	DET
cana-5767	85	9	original	original	ADJ
cana-5767	85	10	and	and	CCONJ
cana-5767	85	11	perturbed	perturb	VERB
cana-5767	85	12	graphs	graph	NOUN
cana-5767	85	13	.	.	PUNCT
cana-5767	86	1	o	o	NOUN
cana-5767	86	2	visualization	visualization	NOUN
cana-5767	86	3	of	of	ADP
cana-5767	86	4	similar	similar	ADJ
cana-5767	86	5	cluster	cluster	NOUN
cana-5767	86	6	structures	structure	NOUN
cana-5767	86	7	.	.	PUNCT
cana-5767	87	1	references	reference	NOUN
cana-5767	87	2	1	1	NUM
cana-5767	87	3	.	.	PUNCT
cana-5767	88	1	piet	piet	PROPN
cana-5767	88	2	van	van	PROPN
cana-5767	88	3	mieghem	mieghem	PROPN
cana-5767	88	4	&	&	CCONJ
cana-5767	88	5	yingyue	yingyue	PROPN
cana-5767	88	6	ke	ke	PROPN
cana-5767	88	7	(	(	PUNCT
cana-5767	88	8	apr	apr	PROPN
cana-5767	88	9	2025	2025	NUM
cana-5767	88	10	)	)	PUNCT
cana-5767	88	11	–	–	PUNCT
cana-5767	88	12	“	"	PUNCT
cana-5767	88	13	some	some	DET
cana-5767	88	14	laplacian	laplacian	ADJ
cana-5767	88	15	eigenvalues	eigenvalue	NOUN
cana-5767	88	16	can	can	AUX
cana-5767	88	17	be	be	AUX
cana-5767	88	18	computed	compute	VERB
cana-5767	88	19	by	by	ADP
cana-5767	88	20	matrix	matrix	NOUN
cana-5767	88	21	perturbation	perturbation	NOUN
cana-5767	88	22	”	"	PUNCT
cana-5767	88	23	2	2	NUM
cana-5767	88	24	.	.	X
cana-5767	88	25	jack	jack	PROPN
cana-5767	88	26	spalding	spalding	PROPN
cana-5767	88	27	-	-	PUNCT
cana-5767	88	28	jamieson	jamieson	PROPN
cana-5767	88	29	(	(	PUNCT
cana-5767	88	30	june	june	PROPN
cana-5767	88	31	2025	2025	NUM
cana-5767	88	32	)	)	PUNCT
cana-5767	88	33	–	–	PUNCT
cana-5767	88	34	“	"	PUNCT
cana-5767	88	35	reweighted	reweighte	VERB
cana-5767	88	36	spectral	spectral	ADJ
cana-5767	88	37	partitioning	partitioning	NOUN
cana-5767	88	38	works	work	NOUN
cana-5767	88	39	:	:	PUNCT
cana-5767	88	40	bounds	bound	VERB
cana-5767	88	41	for	for	ADP
cana-5767	88	42	special	special	ADJ
cana-5767	88	43	graph	graph	NOUN
cana-5767	88	44	classes	class	NOUN
cana-5767	88	45	”	"	PUNCT
cana-5767	88	46	3	3	NUM
cana-5767	88	47	.	.	X
cana-5767	88	48	keshavan	keshavan	PROPN
cana-5767	88	49	,	,	PUNCT
cana-5767	88	50	montanari	montanari	PROPN
cana-5767	88	51	&	&	CCONJ
cana-5767	88	52	oh	oh	PROPN
cana-5767	88	53	(	(	PUNCT
cana-5767	88	54	2024	2024	NUM
cana-5767	88	55	,	,	PUNCT
cana-5767	88	56	ieee	ieee	NOUN
cana-5767	88	57	tit	tit	NOUN
cana-5767	88	58	)	)	PUNCT
cana-5767	88	59	–	–	PUNCT
cana-5767	88	60	“	"	PUNCT
cana-5767	88	61	bias	bias	NOUN
cana-5767	88	62	-	-	PUNCT
cana-5767	88	63	corrected	correct	VERB
cana-5767	88	64	joint	joint	ADJ
cana-5767	88	65	spectral	spectral	ADJ
cana-5767	88	66	embedding	embedding	NOUN
cana-5767	88	67	...	...	PUNCT
cana-5767	88	68	”	"	PUNCT
cana-5767	89	1	4	4	NUM
cana-5767	89	2	.	.	PUNCT
cana-5767	89	3	marianna	marianna	PROPN
cana-5767	89	4	pensky	pensky	PROPN
cana-5767	89	5	(	(	PUNCT
cana-5767	89	6	nov	nov	PROPN
cana-5767	89	7	2024	2024	NUM
cana-5767	89	8	)	)	PUNCT
cana-5767	89	9	–	–	PUNCT
cana-5767	89	10	“	"	PUNCT
cana-5767	89	11	davis	davis	PROPN
cana-5767	89	12	–	–	PUNCT
cana-5767	89	13	kahan	kahan	PROPN
cana-5767	89	14	theorem	theorem	VERB
cana-5767	89	15	in	in	ADP
cana-5767	89	16	the	the	DET
cana-5767	89	17	two	two	NUM
cana-5767	89	18	-	-	PUNCT
cana-5767	89	19	to	to	ADP
cana-5767	89	20	-	-	PUNCT
cana-5767	89	21	infinity	infinity	NOUN
cana-5767	89	22	norm	norm	NOUN
cana-5767	89	23	and	and	CCONJ
cana-5767	89	24	its	its	PRON
cana-5767	89	25	application	application	NOUN
cana-5767	89	26	to	to	PART
cana-5767	89	27	perfect	perfect	ADJ
cana-5767	89	28	clustering	clustering	NOUN
cana-5767	89	29	”	"	PUNCT