Contagions in Random Networks with Overlapping Communities

We consider a threshold epidemic model on a clustered random graph with overlapping communities. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the matrix whose largest eigenvalue gives the phase transition.Comment: Minor modifications for the second version: added comments (end of Section 3.2, beginning of Section 5.3); moved remark (end of Section 3.1, beginning of Section 4.1); corrected typos; changed titl

Impact of Clustering on Diffusions and Contagions in Random Networks

International audienceMotivated by the analysis of social networks, we study a model of network that has both a tunable degree distribution and a tunable clustering coefficient. We compute the asymptotic (as the size of the population tends to infinity) for the number of acquaintances and the clustering for this model. We analyze a contagion model with threshold effects and obtain conditions for the existence of a large cascade. We also analyze a diffusion process with a given probability of contagion. In both cases, we characterize conditions under which a global cascade is possible

How clustering affects epidemics in random networks

International audienceMotivated by the analysis of social networks, we study a model of random networks that has both a given degree distribution and a tunable clustering coefficient. We consider two types of growth process on these graphs that model the spread of new ideas, technologies, viruses, or worms: the diffusion model and the symmetric threshold model. For both models, we characterize conditions under which global cascades are possible and compute their size explicitly, as a function of the degree distribution and the clustering coefficient. Our results are applied to regular or power-law graphs with exponential cutoff and shed new light on the impact of clustering

Analysis of large random graphs

Plusieurs types de réseaux du monde réel peuvent être représentés par des graphes. Comme il s'agit de réseaux de très grande taille, leur topologie détaillée est généralement inconnue, et nous les modélisons par de grands graphes aléatoires ayant les mêmes propriétés statistiques locales que celles des réseaux observés. Un exemple de telle propriété est la présence de regroupements dans les réseaux réels : si deux individus ont un ami en commun, ils ont également tendance à être amis entre eux. Etudier des modèles de graphes aléatoires qui soient à la fois appropriés et faciles à aborder d'un point de vue mathématique représente un challenge, c'est pourquoi nous considérons plusieurs modèles de graphes aléatoires possédant ces propriétés. La propagation d'épidémies dans les graphes aléatoires peut être utilisée pour modéliser plusieurs types de phénomènes présents dans les réseaux réels, comme la propagation de maladies, ou la diffusion d'une nouvelle technologie. Le modèle épidémique que nous considérons dépend du phénomène que nous voulons représenter : . un individu peut contracter une maladie par un simple contact avec un de ses amis (ces contacts étant indépendants), . mais une nouvelle technologie est susceptible d'être adoptée par un individu lorsque beaucoup de ses amis ont déjà la technologie en question. Nous étudions essentiellement ces deux différents cas de figure. Dans chaque cas, nous cherchons à savoir si une faible proportion de la population initialement atteinte (ou ayant la technologie en question) peut propager l'épidémie à une grande partie de la population.Several kinds of real-world networks can be represented by graphs. Since such networks are very large, their detailed topology is generally unknown, and we model them by large random graphs having the same local statistical properties as the observed networks. An example of such properties is the fact that real-world networks are often highly clustered : if two individuals have a friend in common, they are likely to also be each other's friends. Studying random graph models that are both appropriate and tractable from a mathematical point of view is challenging, that is why we consider several clustered random graph models. The spread of epidemics in random graphs can be used to model several kinds of phenomena in real-world networks, as the spread of diseases, or the diffusion of a new technology. The epidemic model we consider depends on the phenomenon we wish to represent : . an individual can contract a disease by a single contact with any of his friends (such contacts being independent), . but a new technology is likely to be adopted by an individual if many of his friends already have the technology in question. We essentially study these two cases. In each case, one wants to know if a small proportion of the population initially infected (or having the technology in question) can propagate the epidemic to a large part of the population.PARIS7-Bibliothèque centrale (751132105) / SudocSudocFranceF