Émilie Coupechoux Résumé En Français

Several kinds of real-world networks can be represented by graphs, in which vertices represent individuals (in social networks), or web pages (in the case of the World Wide Web), to mention only these examples. Each edge of the graph stands for an interaction between vertices: in social networks, an edge is present between two vertices if the individuals they represent are friends; in the World Wide Web, edges represent hyper-links between web pages. 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: if it is the case, we say that a ’cascade’ is possible. The phase transition for this phenomenon is closely related to the appearance of a giant component in a random graph (there is a ’giant component’ in a random graph if the size of its largest connected component is linear with respect to the total size of the graph). In particular, the study of random graphs allows the prediction of global properties (knowing in which case a cascade is possible or not) for large networks on which we have only local data.