Flooding in Weighted Random Graphs

Marc Lelarge

Description

Niveau: Supérieur, Doctorat, Bac+8
Flooding in Weighted Random Graphs Hamed Amini? Moez Draief† Marc Lelarge‡ Abstract In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of n, when the size of the graph n tends to infinity. We also derive the exact value for the prefactors. These allow us to analyze an asynchronous random- ized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average. 1 Introduction Driven by the distributed nature of modern network architectures, there has been intense research to devise algorithms to ensure effective network computation. Of particular interest is the problem of global node outreach, whereby some major event happening in one part of the network has to be communicated to all other nodes. In this context, gossip protocols have been identified as simple, efficient and robust mechanisms for disseminating and retrieving information for various network topologies.

• random graphs


• uniformly chosen


• typical weighted


• ing poisson


• flood- ing time


• weighted diameter


• path connecting


• has degree

Sujets

• Random graph

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 56
• Langue: English

Extrait

Flooding in Weighted Random Graphs Hamed Amini ∗ Moez Draief † Marc Lelarge ‡

Abstract evaluated in terms of the time it takes to complete. In this paper, we study the impact of edge weights on  This in particular depends on the underlying network distancesindilutedrandomgraphs.Weinterpretthese t d o i p ffe o r lo e g nt y, v n e a rt m i e c l e y s t o h f e t e h x e ist n e e n t ceof k. sho I rt n p p a r t a h c s t between weightsasdelays,andtakethemasi.i.dexponential mayimaginethatthereareo w th or erparameters, ic b e e , s o d n es e randomvariables.Weanalyzetheweightedflooding henetworktopology,tobetakenintoaccounts i uch time defined as the minimum time needed to reach all  t nodesfromoneuniformlychosennode,andtheweighted a e s xa the le communicationdela hi y s s co b n e t t e w x e t e , n th n e o s d p e r s ea d d u o e ft f h or e diametercorrespondingtothelargestdistancebetween infor m m p ati t o o n c i o n ng t e h s e ti n o e n t . w I o n rk t canbethohtofasafluid anypairofvertices.Undersomeregularityconditions penetratingthenetwokminiscento u f g theproblemof onthedegreesequenceoftherandomgraph,weshow fiercolatio r ni r n e arandommedium.Inthis that these quantities grow as the logarithm of n ,when rst-passag w e ill p consideranasynchronousmodelinwhich the size of the graph n tendstoinfinity.Wealsoderive pa c p h er, ed w g e eofthenetworkisequippedwitharandom the exact value for the prefactors.  ea These allow us to analyze an asynchronous random-  delay modeled by an exponential random variable with izedbroadcastalgorithmforrandomregulargraphs. mean O one. fthemainmotivationsofourworkcomes Ourresultsshowthattheasynchronousversionofthe fromp n e e e o rnetworks.Inparticular,tomotivate algorithmperformsbetterthanitssynchronizedversion: ourrand r o -t m o- g p r ee hodel,werecallthatthemostrel- in the large size limit of the graph, it will reach the whole  ap m network faster even if the local dynamics are similar on  evant properties of peer-to-peer networks are connec- average. tivity, small average degree, and approximate regular- ity of the degrees of the vertices. The random graph 1 Introduction model considered in this paper, explained in detail in Section 2, has these properties, and covers the classi- Driven by the distributed nature of modern network  cal G ( n r ) model, which is the random graph model in architectures, there has been intense research to devise algorithmstoensureeffectivenetworkcomputation. s w e h t ic o h f a n - g v r e a rt p e h x i r s -r d e r g a u w la n r u g n r i a fo p r h m s, ly w a h t er r e an r d i o s m a f c r o o n m sta th n e t Of particular interest is the problem of global node outreach, whereby some major event happening in one  not depending on n . For this model of networks, we partofthenetworkhastobecommunicatedtoall i c n o i n ti si a d ll e y r , t o h n e e p o u f s t h he mo n d o e d l e f s or ob d t is a s i e ns mi s n o a m ti e ng pi i e n c f e or of m i a n ti f o or n - : other nodes. In this context, gossip protocols have been  mation. Then every node which has the information identified as simple, efficient and robust mechanisms fordisseminatingandretrievinginformationforvarious p T a h s e se c s la i s t si t c o al an m o o t d h e e l r g n o o e d s e it c e h ra os ti e v n el a y m a o n n d g a i l t l s n n o e d ig es hb h o a r v s e . network topologies. These mechanisms rely on simple  the same clock In each successive round, the nodes periodic local operations between neighboring nodes  . [18]. f h o a r v m in ly g a t t h r e a i n n d f o or m m t a h ti e o n n ei c g h h o b o o s r e t i h n e d y ep tr e a n n d s e m nt i l t y to a . n I d n u th n i i- s Flooding corresponds to the most commonly used such process: a source node that first records the event  paper, we analyze an asynchronous randomized broad- notifies all the nodes within its reach. Subsequently,  cast algorithm. Namely nodes are not anymore assumed eachoftheseneighborsforwardsinformationtoallof t d o en b t e P s o y i n ss c o hr n o c n l i o z c e k d . ,s A ot n h o a d t e e r a e c c h ei n v o i d n e g h t a h s e a in n fo i r nd m e a p ti e o n n - its neighbors and so on. If the underlying network isconnectedsuchinformationwilleventuallyreachall o w w ill n t c r l a o n c s k m . it W it h t e o n a th r e an g d r o a m ph ne o i f g n h e b i o g r hb at or e s ac is h a tic r k an o d f o it m s the nodes. The performance of this procedure can be r -regular graph, we show that the asynchronous version of the algorithm performs better than its synchronized ∗ INRIA-ENS, [e-mail] version (see Section 3). To the best of our knowledge, † Imperial College London, [e-mail] ‡ INRIA-ENS, [e-mail] our work is the first to study this model in an asyn-