William-Final.DVI


Mathematical Problems of Computer Science 46, 126{131, 2016.

Gossiping P r oper ties of the M odi¯ed KnÄodel Gr aphs

V ilya m H . H o vn a n ya n

Institute for Informatics and Automation Problems of NAS RA

williamhovnanyan@gmail.com

Abstract

In this paper we consider the gossiping process implemented on several modi¯ca-
tions of KnÄodel graphs. We show the ability of KnÄodel graphs to remain good network
topology for gossiping even in case of cyclic permutation of its edge weights. The re-
sults shown in this paper could help us to construct edge-disjoint paths between any
pairs of vertices of the KnÄodel graph.

Keywords: Graphs, Networks, Telephone problem, Gossip problem, KnÄodel
graphs.

1 . In t r o d u c t io n

Go s s ip in g is o n e o f t h e b a s ic p r o b le m s o f in fo r m a t io n d is s e m in a t io n in c o m m u n ic a t io n n e t -
wo r ks . Th e g o s s ip p r o b le m ( a ls o kn o wn a s a t e le p h o n e p r o b le m ) is a t t r ib u t e d t o A . B o yd ( s e e
e .g ., [4 ] fo r r e vie w) , a lt h o u g h t o t h e b e s t kn o wle d g e o f t h e r e vie we r s , it wa s ¯ r s t fo r m u la t e d
b y R . Ch e s t e r s a n d S . S ilve r m a n ( U n iv. o f W it wa t e r s r a n d , u n p u b lis h e d , 1 9 7 0 ) . Co n s id e r
a s e t o f n p e r s o n s ( n o d e s ) e a c h o f wh ic h in it ia lly kn o ws s o m e u n iqu e p ie c e o f in fo r m a t io n
t h a t is u n kn o wn t o t h e o t h e r s , a n d t h e y c a n m a ke a s e qu e n c e o f t e le p h o n e c a lls t o s p r e a d
t h e in fo r m a t io n . D u r in g a c a ll b e t we e n t h e g ive n t wo n o d e s , t h e y e xc h a n g e t h e wh o le in -
fo r m a t io n kn o wn t o t h e m a t t h a t m o m e n t . Th e p r o b le m is t o ¯ n d a s e qu e n c e o f c a lls wit h
a m in im u m le n g t h ( m in im a l g o s s ip s c h e m e ) , b y wh ic h a ll t h e n o d e s will kn o w a ll p ie c e s o f
t h e in fo r m a t io n ( c o m p le t e g o s s ip in g ) . It h a s b e e n s h o wn in n u m e r o u s wo r ks [1 ]{ [4 ] t h a t t h e
m in im a l n u m b e r o f c a lls is 2 n ¡ 4 wh e n n ¸ 4 a n d 1 , 3 fo r n = 2 ; 3 , r e s p e c t ive ly. S in c e t h e n
m a n y va r ia t io n s o f g o s s ip p r o b le m h a ve b e e n in t r o d u c e d a n d in ve s t ig a t e d ( s e e e .g ., [5 ]{ [1 0 ],
[1 3 ]{ [1 7 ]) .

A n o t h e r va r ia n t o f Go s s ip p r o b le m c a n b e fo r m u la t e d b y c o n s id e r in g t h e m in im u m
a m o u n t o f t im e r e qu ir e d t o c o m p le t e g o s s ip in g a m o n g n p e r s o n s , wh e r e t h e c a lls b e t we e n
t h e n o n -o ve r la p p in g p a ir s o f n o d e s c a n t a ke p la c e s im u lt a n e o u s ly a n d e a c h c a ll r e qu ir e s o n e
u n it o f t im e ( [1 1 ], [1 2 ],[1 8 ]) .

Ob vio u s ly, t h e g o s s ip p r o b le m c a n b e e a s ily m o d e le d a s a g r a p h , t h e ve r t ic e s o f wh ic h
r e p r e s e n t p e o p le in g o s s ip s c h e m e a n d e d g e s r e p r e s e n t c a lls b e t we e n t h e m ( e a c h o f t h e m
h a s we ig h t wh ic h r e p r e s e n t s t h e m o m e n t wh e n c o m m u n ic a t io n t o o k p la c e ) . S o t h e g r a p h is
c a lle d a c o m p le t e g o s s ip g r a p h if t h e r e a r e a s c e n d in g p a t h s b e t we e n a ll t h e p a ir s o f ve r t ic e s
o f t h is g r a p h . A p a t h b e t we e n t wo ve r t ic e s is a n a s c e n d in g p a t h if e a c h n e xt e d g e h a s h ig h e r
we ig h t t h a n t h e p r e vio u s o n e .

1 2 6



V. Hovnanyan 1 2 7

K n Äo d e l g r a p h s a r e o n e o f t h e 3 we ll-kn o wn n e t wo r k t o p o lo g ie s fo r g o s s ip in g / b r o a d c a s t in g
( a lo n g s id e wit h h yp e r c u b e a n d r e c u r s ive c ir c u la n t g r a p h s ) .

De¯nition 1: The K nÄodel graph on n ¸ 2 vertices (n even) and of maximum degree ¢ ¸ 1
is denoted by W¢ ;n. The vertices of W¢ ;n are the pairs ( i; j ) with i = 1 ; 2 and 0 · j · n=2 ¡ 1 .
F or every j, 0 · j · n=2 ¡ 1 and l = 1 ; : : : ; ¢ , there is an edge with the label (weight) l
between the vertex ( 1 ; j ) and ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n=2 ) . The edges with the given label l are
said to be in dimension l.

N o t e t h a t W1;n c o n s is t s o f n= 2 d is c o n n e c t e d e d g e s . Fo r ¢ ¸ 2 , W¢ ;n is c o n n e c t e d .
In t h is p a p e r we g ive s o m e n e w o b s e r va t io n s , m o s t ly in t e r m s o f g o s s ip in g , c o n c e r n in g

a s lig h t m o d ī c a t io n o f K n Äo d e l g r a p h s , o b t a in e d b y a c yc lic p e r m u t a t io n o f t h e d im e n s io n s
o f e d g e s . Fo r t h e g r a p h u n d e r c o n s id e r a t io n , t h e m o d i¯ c a t io n lo o ks a s fo llo ws : d im e n s io n
o f t h e e d g e c o n n e c t in g t h e ve r t ic e s ( 1 ; j ) a n d ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n=2 ) , is r e p la c e d b y
( l + p ) m o d dlo g 2 ne wh e r e p = 1 ; : : : ; d¢ e. W e ¯ r s t c o n s id e r t h e c a s e o f n = 2 k ¡ 2 a n d s h o w
t h e a b ilit y o f K n Äo d e l g r a p h s t o p r e s e r ve g o s s ip in g p r o p e r t ie s fo r t h e o p e r a t io n d e s c r ib e d .
Th e n we t r y t o g e n e r a liz e t h e a b o ve s t a t e m e n t a b o u t K n Äo d e l g r a p h s fo r a ll g e n e r ic n 6= 2 k.
A t t h e e n d we s h o w t h e in a b ilit y o f t h e s e g r a p h s t o p r e s e r ve g o s s ip in g p r o p e r t ie s in t h e c a s e
o f n = 2 k, e xc e p t fo r o n e s p e c ī c c a s e .

2 . Mo d ī e d K n Äo d e l Gr a p h s W it h n = 2 k ¡ 2 V e r t ic e s
In t h is s e c t io n we a r e g o in g t o c o n s id e r t h e m o d i¯ e d K n Äo d e l g r a p h s wit h t h e n u m b e r o f
ve r t ic e s e qu a l t o 2 k ¡ 2 , wh e r e k is a n y in t e g e r wit h k ¸ 3 . Th e d e ¯ n it io n o f m o d i¯ e d
K n Äo d e l g r a p h is a s fo llo ws :

De¯nition 2: The modi¯ed K nÄodel graph on n ¸ 2 vertices (n even) and of maximum degree
¢ ¸ 1 is denoted by M¢ ;n ( p) . The vertices of M¢ ;n ( p ) are the pairs ( i; j ) with i = 1 ; 2 and
0 · j · n= 2 ¡ 1 , respectively. F or every j, 0 · j · n=2 ¡ 1 and l = 1 ; : : : ; ¢ , there is an edge
with the label ( l + p ) m o d dlo g 2 ne between the vertices ( 1 ; j ) and ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n= 2 )
where p is a ¯xed integer for that graph with possible values p = 1 ; : : : ; ¢ . The edges with
the given label l + p are said to be of dimension l + p.

A c c o r d in g t o t h e a b o ve d e ¯ n it io n , t h e r e e xis t ¢ ¡ 1 m o d i¯ c a t io n s fo r e a c h W¢ ;n. In
t h is p a p e r we c o n s id e r o n ly t h e c a s e , wh e n ¢ = dlo g 2 ne, a n d t h is va lu e is a s s u m e d fo r a ll
r e fe r e n c e s o f ¢ o n wa r d s .

De¯nition 3: Consider two graphs G1 = ( V; E1 ) and G2 = ( V; E2 ) with the same set of
vertices V and labeled edge sets E1 and E2, respectively. The edge sum of these graphs is
a graph G1 + G2 = G = ( V; E ) with E = E1 [ E2, whose edges e 2 E are labeled by the
following rules:

tG ( e ) =

8
<
:

tG1 ( e) ; if e 2 E1;
tG2 ( e) + m a x

e02E1
tG1 ( e

0 ) ; if e 2 E2: ( 1 )

where tG ( e) is tha label of the edge e in the resulting graph G.

It is kn o wn t h a t if G = W¢ ;n + W1;n, t h e n G is a c o m p le t e g o s s ip g r a p h .
L e t u s ¯ r s t c o n s id e r t h e g r a p h G0 = M¢ ;n ( ¢ ) + M1;n ( ¢ ) .

T heor em 1: The graph G0 is a complete gossip graph.



1 2 8 Gossiping Properties of the Modi¯ed KnÄodel Graphs

0

t h a t in c lu d e s t h e ve r t ic e s ( i; j ) , wh e r e i = 1 ; 2 a n d 1 · j · ¢ . L e t u s p a r t ic u la r ly fo c u s o n
t h e ve r t ic e s , fo r wh ic h i = 0 a n d 1 · j · ¢ , a n d c o n s id e r t h e c a ll in d im e n s io n 1 . Th e r e is
a n e d g e in d im e n s io n 1 b e t we e n t h e s e a n d ( 1 ; j0 ) ve r t ic e s , wh e r e 2 ¢ · j0 · n=2 . Th e s a m e is
t r u e a b o u t t h e ve r t ic e s ( 1 ; j ) a n d ( 0 ; j0 ) , wh e r e 1 · j · 2 ¢ a n d 2 ¢ < j0 · n=2 . It is o b vio u s
t h a t t h e s e e d g e s c o n n e c t a ll t h e r e m a in in g 2 k ¡ 2 ¡ 2 ¢ = 2 ¢ +1 ¡ 2 ¢ ¡ 2 = 2 ¢ ¡ 2 ve r t ic e s
t o t h e SUBG0 . On t h e o t h e r h a n d , we h a ve t h a t SUBG0 is a t r e e r o o t e d a t t h e ve r t e x ( 0 ; 1 ) .
Th u s , t h e r e e xis t a s c e n d in g p a t h s b e t we e n e ve r y ve r t e x o f SUBG0 a n d t h e ve r t e x ( 0 ; 1 ) .
Co n s id e r in g t h e fa c t t h a t e a c h e d g e in t h e SU BG0 h a s d im e n s io n g r e a t e r t h a n 1 , it fo llo ws
t h a t t h e r e e xis t s a n a s c e n d in g p a t h b e t we e n a n y o t h e r ve r t e x a n d t h e ve r t e x ( 0 ; 1 ) . Th e t wo
fa c t s t h a t t h e c h o ic e o f t h e ve r t e x ( 0 ; 1 ) wa s a r b it r a r y, a ls o t h e m o d i¯ e d K n Äo d e l g r a p h b e in g
s ym m e t r ic b y c o n s t r u c t io n , p o in t o u t t o c o m p le t e g o s s ip in g o f G0.

Cor ollar y 2: Since the graph G0 is a gossip graph, we can claim that M¢ ;n ( p) , where 0 ·
p · ¢ , is also a gossip graph.

t o a d d W1;n in o r d e r t o o b t a in a c o m p le t e g o s s ip g r a p h .

3 . Mo d ī e d K n Äo d e l Gr a p h s W it h n = 2 k V e r t ic e s

In t h is s e c t io n we c o n s id e r m o d i¯ e d K n Äo d e l g r a p h s wit h 2 k ve r t ic e s . K n Äo d e l g r a p h s wit h 2 k

ve r t ic e s a r e d i®e r e n t in t e r m s o f g o s s ip in g s in c e W¢ ;n is a c t u a lly a g o s s ip g r a p h . H e n c e , in
t h is c a s e t h e r e is n o n e e d t o a d d W1;n t o g e t a c o m p le t e g o s s ip g r a p h .

To s t a r t wit h , le t u s c o n s id e r M¢ ;n ( 2 ) .

T heor em 3: The graph M¢ ;n ( 2 ) is a complete gossip graph.

in c a s e j is e ve n , a n d jnew = n=2 + bj=2 c in c a s e o f o d d j.
W e o b t a in t wo c o m p o n e n t s t h a t a r e K n Äo d e l g r a p h s wit h t h e n u m b e r o f ve r t ic e s e qu a l t o

n=2 , a n d t h e s e c o m p o n e n t s a r e c o n n e c t e d b y t h e e d g e s o f la b e l ¢ . Th e r e fo r e , if we s im p ly
r e m o ve t h e s e e d g e s , t h e n t h e r e will b e t wo d is jo in t c o m p o n e n t s wh ic h a r e K n Äo d e l g r a p h s
t h e m s e lve s . A n d t h e r e is a n o n e -t o -o n e c a ll m a p p in g b e t we e n t h e s e t wo c o m p o n e n t s wit h
t h e c a lls la b e le d b y lo g 2 n ( h ig h e s t la b e le d e d g e s ) . Ob vio u s ly, t h is g r a p h is a c o m p le t e g o s s ip
g r a p h .

T heor em 4: The graph M¢ ;n ( p ) , where p 6= 2 , is not a gossip graph.

P r oof: S in c e t h is g r a p h is s ym m e t r ic , it is e n o u g h t o s h o w t h a t t h e r e e xis t s a n a s c e n d in g
p a t h fr o m a n y o t h e r ve r t e x t o t h e ve r t e x ( 0 ; 1 ) . L e t u s c o n s id e r t h e s u b g r a p h o f G0, SUBG

P r oof. It is a n e a s y e xe r c is e t o ve r ify t h a t G0 is is o m o r p h ic t o W¢ ;n ( fo r d e t a ile d p r o o f
r e fe r t o [2 3 ]) . H e n c e , we c a n r e p e a t t h is p r o c e d u r e , a n d e a c h t im e we will o b t a in a n e w
M¢ ;n ( p) , p = ¢ ¡ 1 ; ¢ ¡ 2 ; :::; 1 , t h a t is is o m o r p h ic t o in it ia l W¢ ;n. Th u s , t h e r e is n o n e e d

P r oof: L e t 's s t a r t o u r p r o o f b y d o in g t h e fo llo win g t r a n s fo r m a t io n o n t h e ve r t e x s e t o f
t h is g r a p h V . Th e e le m e n t s o f t h is s e t a r e t h e p a ir s ( i; j ) . In c a s e o f i = 1 , if j is o d d , t h e n

jnew = dj= 2 e a n d jnew = n=2 + j=2 , o t h e r wis e . If i = 2 , t h e r u le s a r e a s fo llo ws : jnew = j=2



V. Hovnanyan 1 2 9

L e t u s d e n o t e t h e s u b s e t o f t h e e d g e s e t o f g r a p h M wit h t h e we ig h t wfix = log2n¡i+2 b y
E0, a n d it s e le m e n t s b y e0l, wh e r e l = 1 ; 2 ; : : : n= 2 . Th e o p e r a t io n A

¡
e0
l
( s e e [2 2 ]) s h o u ld a p p ly

t h e " p e r m u t e lo we r " o p e r a t io n o n a ll t h e e d g e s o f E0 r e s u lt in g in a c o m p le t ly n e w g r a p h .
Th e r e wo u ld o b vio u s ly b e d u p lic a t e ( d o u b le ) e d g e s in t h is g r a p h , s in c e t h e e d g e s wit h we ig h t s
wf ix ¡ 1 a n d wf ix + 1 wo u ld c o n n e c t t h e s a m e ve r t ic e s . H e n c e , t a kin g in t o c o n s id e r a t io n
t h e fa c t t h a t t h e g o s s ip in g t im e , i.e ., t h e n u m b e r o f r o u n d s r e qu ir e d t o c o m p le t e g o s s ip in g
in W¢ ;n wa s log2n ( wh ic h is t h e m o s t o p t im a l va lu e o f t h is p r o p e r t y) , t h e g r a p h M¢ ;n ( p ) is
p r o ve d t o b e a n o n -g o s s ip g r a p h .

A c kn o wle d g e m e n t s

Th e a u t h o r is g r a t e fu l t o P r o f. Y u .H . S h o u ko u r ia n , D r . S . P o g h o s ya n a n d D r . V . P o g h o s ya n
fo r im p o r t a n t d is c u s s io n s a n d c r it ic a l r e m a r ks a t a ll s t a g e s o f t h e wo r k.

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1 3 0 Gossiping Properties of the Modi¯ed KnÄodel Graphs

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[2 0 ] A . S e r e s s , \ Qu ic k Go s s ip in g wit h o u t d u p lic a t e t r a n s m is s io n s " , Graphs and Combina-
torics, vo l. 2 , p p . 3 6 3 -3 8 1 , 1 9 8 6 .

[2 1 ] P . Fr a ig n ia u d , A . L . L ie s t m a n a n d D . S o it e a u , \ Op e n p r o b le m s " , P arallel P rocessing
L etters, vo l. 3 , n o . 4 , p p . 5 0 7 -5 2 4 , 1 9 9 3 .

[2 2 ] V . H . H o vn a n ya n , S u . S . P o g h o s ya n a n d V . S . P o g h o s ya n , \ Me t h o d o f lo c a l in t e r c h a n g e
t o in ve s t ig a t e Go s s ip p r o b le m s " , Transactions of IIAP of NAS R A, M athematical P rob-
lems of Computer Science, vo l. 4 0 , p p . 5 -1 2 , 2 0 1 3 .

[2 3 ] L .H . K h a c h a t r ia n , H .S . H a r u t o u n ia n , \ On Op t im a l B r o a d c a s t Gr a p h s " , P roc. of the
fourth Int. Colloquium on Coding Theory, 6 5 -7 2 , A r m e n ia , 1 9 9 1 .

Submitted 28.07.2016, accepted 02.11.2016.

Øá¹ÇýÇϳóí³Í KnÄodel ·ñ³ýÝ»ñÇ gossiping
(µ³Ùµ³ë³Ýù³ÛÇÝ) ѳïÏáõÃÛáõÝÝ»ñÁ

ì. ÐáíݳÝÛ³Ý

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Çñ³Ï³Ý³óíáÕ gossiping åñáó»ëÁ: òáõÛó ¿ ïñíáõÙ K n Äo d e l ·ñ³ýÇ gossiping-Ç Ñ³Ù³ñ
ɳí ó³Ýó³ÛÇÝ ïáåáÉá·Ç³ ѳݹÇë³Ý³Éáõ ϳñáÕáõÃÛáõÝÁ ÝáõÛÝÇëÏ ¹ñ³ ÏáÕ»ñÇ ÏßÇéÝ»ñÇ
óÇÏÉÇÏ í»ñ³¹³ë³íáñÙ³Ý å³ñ³·³ÛáõÙ: Ðá¹í³ÍáõÙ óáõÛó ïñí³Í ³ñ¹ÛáõÝùÝ»ñÁ ϳñáÕ
»Ý û·ï³Ï³ñ ÉÇÝ»É K n Äo d e l ·ñ³ýÇ Ï³Ù³Û³Ï³Ý »ñÏáõ ·³·³ÃÝ»ñÇ ÙÇç¨ ÏáÕ³ÝϳË
׳ݳå³ñÑÝ»ñÇ Ï³éáõóÙ³Ý Ñ³Ù³ñ:



V. Hovnanyan 1 3 1

Ãîññèï ñâîéñòâà ìîäèôèöèðîâàííûõ Êíåäåë ãðàôîâ
Â. Îâíàíÿí

Àííîòàöèÿ

 ýòîé ñòàòüå ìû ðàññìàòðèâàåì ïðîöåññ ãîññèïà ðåàëèçîâàííûé íà
íåêîòîðûõ ìîäèôèêàöèÿõ Êíåäåë ãðàôîâ. Ïîêàçàíà ñïîñîáíîñòü Êíåäåë ãðàôîâ
îñòàâàòüñÿ õîðîøåé ñåòåâîé òîïîëîãèåé äëÿ ãîññèïà äàæå â ñëó÷àå öèêëè÷åñêîé
ïåðåñòàíîâêè âåñîâ ðåáåð. Ðåçóëüòàòû, ïðåäñòàâëåííûå â äàííîé ðàáîòå
ñïîñîáñòâóþò ïîñòðîåíèþ ðåáåðíî-íåïåðåñåêàþùèõñÿ ïóòåé ìåæäó ëþáûìè
ïàðàìè âåðøèí Êíåäåë ãðàôîâ.