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"Epidemics and Percolation in Random Networks"

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THESE` DE DOCTORAT pr´esent´eepar Hamed Amini Sp´ecialit´e: Mathematiques´ Appliquees´ Equipe´ d'acceuil : Ecole´ Normale Sup´erieure- INRIA Rocquencourt (Projet TREC) Epidemics and Percolation in Random Networks soutenue le 24 Juin 2011, devant le jury compos´ede: Venkat ANANTHARAM University of California, Berkeley Examinateur Fran¸coisBACCELLI INRIA - Ecole´ Normale Sup´erieure Directeur de th`ese Francis COMETS Universit´eParis 7 Examinateur Remco van der HOFSTAD Eindhoven University of Technology Rapporteur Marc LELARGE INRIA - Ecole´ Normale Sup´erieure Directeur de th`ese Laurent MASSOULIE´ Technicolor Research Rapporteur Gilles SCHAEFFER Ecole´ Polytechnique Examinateur To my family Remerciements Ma reconnaissance revient principalement `ames directeurs de th`eseFrancois Baccelli et Marc Lelarge qui, au cours de ces quatre ann´ees,ont encadr´emon travail sur un sujet passionnant. Je les remercie pour leur soutien et leur conseils judicieux. Merci de m'avoir fait d´ecouvrirle monde de la recherche. Je tiens `aremercier Remco van der Hofstad et Laurent Massouli´equi ont eu la lourde t^ache de rapporteurs. Je les remercie infiniment pour la lecture et la correction de mon manuscrit et pour l'int´er^etqu'ils ont port`e`amon travail. Je remercie ´egalement les autres membres du jury, Venkat Anantharam, Francis Comets et Gilles Schaeffer de m'avoir fait l'honneur de participer `amon jury de th`ese. I would like to thank several people for the privilege to visit their group and work with them during my thesis. First, Moez Draief at Imperial College in London, then Nikolaos Fountoulakis at Max-Planck-Institut in Saarbr¨cken, and finally Yuval Peres for visiting his group at Microsoft research in Redmond. I'm grateful for fruitful discussion with the members of these respective groups. L'application en finance d'une partie de mes travaux n'aurait pas ´et´epossible sans l'apport de Rama Cont et Andreea Minca, `aqui je remercie ´egalement. J'ai eu la chance de passer ma th`eseau projet TREC o`uj'ai beaucoup appr´eci´ela grande lib- ert´eofferte pour la recherche, dans une ambiance stimulante et chaleureuse. Je tiens `aremercier tous les coll`eguesdu projet TREC et en particulier Ana, Anne, Bartek, Bruno, Calvin, Emilie, Furcy, Fran¸cois-Xavier, Fr´ed´eric,Florence, Giovanna, Justin, Mathieu, Minh-ahn, Nadir, Omid, Pierre, Tien, Van Minh et Yogesh. Je remercie ´egalement Nathalie Abiola, Claudette Dessertaine, Christine Ferret, Joelle Is- nard, Audrey Lemarechal et H´el`neMilome pour leur gentillesse et leur aide dans les t^aches administratives. Les derni`ereslignes de ces remerciements vont `ames parents, ma soeur Hadis, `aPooran et `aAndreea qui mont toujours soutenu durant cette th`ese.Enfin, je remercie mille fois mon fr`ere Omid pour son soutien et sa pr´esenceau moment difficile. Preface L'omnipr´esencedes r´eseauxdans tous les aspects de la vie moderne et les probl`emesassoci´es ont incit´ede nombreuses recherches dans le but de comprendre et pr´edirele comportement de telles structures irr´eguli`ers,complexes, et ´evolutives. Les exemples de r´eseauxvont de l'Internet jusqu'aux interconnexions d'agents financiers ou bien aux r´eseauxneuronaux. Dot´ede grande capacit´ede calcul et de la disponibilit´ede larges ensembles de donn´ees sur des r´eseauxr´eels(comme l'Internet, les r´eseauxde transport, r´eseauxde t´el´ephone,r´eseaux coauteurs scientifiques, r´eseauxprot´eine-prot´einebiologiques, etc.), l'´etudede r´eseauxcomplexes a ´emerg´ecomme un domaine `acroissance rapide. La plupart des r´eseauxr´eelssont tr`esgrands, de plusieurs milliers de noeuds dans un r´eseau d'entreprises `aplusieurs milliards de noeuds dans certains syst`emesbiologiques. En raison de leur complexit´e,leurs relations ne peuvent ^etred´ecritesqu'en termes statistiques. L'une des observations les plus frappantes est la suivante : des r´eseauxavec des fonctions compl`etement diff´erentes, comme par exemple des r´eseauxsociaux et des r´eseauxbiologiques, partagent des caract´eristiquescommunes. En particulier, nombre de ces syst`emessont des "petits-mondes", ce qui traduit le fait que la distance topologique moyenne dans le r´eseau(qui mesure le nombre moyen de liens `afranchir sur le r´eseaupour aller d'un site `aun autre) varie tr`eslentement avec le nombre total de sites (typiquement comme un logarithme). Une autre d´ecouverte particuli`erement importante est le fait que la fr´equenced'apparition de sites avec k voisins (sites de degr´e k) est une distribution en loi de puissance : si on d´efinitla dis- tribution des degr´espar pk (probabilit´equ'un noeud choisi uniform´ement parmi tous les noeuds −γ ait k voisins), il existe un param`etre γ tel que pk ∼ k . Ce r´esultata permis l'identification d'une nouvelle cat´egoriede r´eseauxdits "sans-´echelle", et diff´erencieces r´eseauxdu graphe clas- sique al´eatoirepropos´epar Erd}oset R´enyi dans les ann´ees60, dans lequel la distribution est ii Poisson et donc homog`ene,dans le sens o`ule nombre de voisins de chaque noeud fluctue tr`es peu autour d'une valeur moyenne. Une question essentielle dans l'´etudedes r´eseauxest l’influence de la topologie d'un r´eseausur sa r´eponse aux facteurs perturbatifs. Par exemple dans de nombreux domaines, il est essentiel de comprendre et caract´eriserla robustesse des r´eseaux,quand on supprime un certain nombre de noeuds, ou de liens. Un autre exemple consiste `a´etudierla propagation des ´epid´emiessur les diff´erents types de r´eseaux.Les ´epid´emiessur des graphes permettent de mod´eliserde nombreux ph´enom`enesdans les r´eseauxtels que la propagation de virus, de vers, de rumeurs, et aussi les faillites des agents financiers, pour en citer quelques-un. Les syst`emesde particules en interaction (comme le processus de contact) ont permis de mod´eliseravec succ`esde tels ph´enom`enes.Ces syst`emesal´eatoiresont deux caract´eristiquesprincipales : ils mod´elisent un grand nombre de particules sur un graphe, et l'´etatde chaque particule d´epend des ´etatsdes particules voisines sur le graphe. Ce genre de syst`emespr´esente g´en´eralement une transition de phase entre une phase active o`ul'´epid´emieinfecte une large proportion du r´eseauet une phase inactive o`ula contagion reste contenue. Le param`etrede contr^olepermettant de changer de phase est la transmissibilit´e de l'infection, et on s'int´eresseg´en´eralement `ala d´eterminationde sa valeur critique. De nouveaux mod`elesmath´ematiques,visant `acapturer les propri´et´esdes syst`emesr´eels organis´esen r´eseaux,ont ´et´ed´evelopp´es.Ces mod`elesont enrichi la th´eoriedes graphes al´eatoires par l'´etudemath´ematiquerigoureuse des lois r´egissant l'´evolution de ces syst`emes,dans le m^eme esprit que dans les graphes d'Erd}os-R´enyi. Les r´eseauxcomplexes ont une structure inhomog`ene, d´ecoulant non seulement du fait que les caract´eristiquesstructurelles des sommets peuvent s'´ecarterfortement de la moyenne, mais aussi du fait que les propri´et´esstatistiques peuvent changer entre les diff´erentes parties du r´eseau. Des ph´enom`enescritiques comme la transition de phase induite par l'apparition d'une composante g´eante ont ´et´ebien ´etudi´essur les graphes d'Erd}os-R´enyi. Des progr`esr´ecents ont ´et´er´ealis´essur des graphes al´eatoiresplus r´ealistestels que les graphe al´eatoiresavec la distribution des degr´esdonn´ee(aussi connu comme le mod`ele de configuration) : e.g. l'´emergencede la composante g´eante, l'existence et les propri´et´esdu k-core, la percolation et les ´epid´emies. Voici une description rapide du contenu de cette th`ese. Chapitre 1. R´eseaux complexes et graphes al´eatoires. Dans ce chapitre, nous introduisons la th´eoriedes graphes al´eatoireset les r´eseauxcomplexes. Nous rappelons d'abord quelques iii notions et r´esultatsde la th´eoriedes processus de branchement, pertinentes pour pr´edireles propri´et´esdes graphes al´eatoires. Nous donnons ensuite les r´esultatsconnus sur l'´emergence d'une composante g´eante, k-core, et les distances dans les graphes d'Erd}os-R´enyi, et finissons le chapitre par rappeler des r´esultatsconnus sur le mod´elede configuration. Chapitre 2. Percolation de premier passage, l'inondation et le diam`etre. Dans ce chapitre, nous consid´eronsl'impact des poids sur les distances dans les graphes al´eatoiresdilu´es. Nous interpr´etonsces poids comme des retards, et les prenons comme des variables al´eatoiresexponen- tielles i.i.d.. Nous analysons le temps d'inondation d´efinicomme le temps minimum n´ecessaire pour atteindre tous les noeuds `apartir d'un noeud choisi d'une mani`ereuniforme, et le diam`etre correspondant au pire cas pour le temps d'inondations. Sous certaines conditions de r´egularit´e sur la s´equencede degr´esdu graphe al´eatoire,nous montrons que ces quantit´escroissent comme le logarithme de n, lorsque la taille du graphe n tend vers l’infini. Nous trouvons ´egalement la valeur exacte des pr´efacteurs. Ce resultat nous permet d'analyser un algorithme de transmis- sion asynchrone dans les graphes al´eatoiresr´eguliers.Nous montrons que la version asynchrone de l'algorithme est plus performante que sa version synchronis´eequand la taille du graphe est suffisament grande, il atteindra l'ensemble du r´eseauplus rapidement, m^emesi le dynamique local est similaire en moyenne. Ce chapitre est fond´esur les articles [4, 7], en collaboration avec M. Draief et M. Lelarge. Chapitre 3. Percolation bootstrap, diffusion et cascades. Dans ce chapitre, nous ´etudions la diffusion et les cascades dans les graphes al´eatoires. Notre mod`elede diffusion peut ^etre consid´er´ecomme une variante d'un processus de croissance d'un automate cellulaire : supposons que chaque site puisse ^etredans l'un des deux ´etatspossibles, inactif ou actif.
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