Some implications of local weak convergence for sparse random graphs

Justin Salez 1
1 TREC - Theory of networks and communications
ENS Paris - École normale supérieure - Paris, INRIA, DI-ENS - Département d'informatique de l'École normale supérieure
Abstract : In the so-called sparse regime where the numbers of edges and vertices tend to infinity in a comparable way, the asymptotic behavior of many graph invariants is expected to depend only upon local statistics. This heuristic originates from the thermodynamic study of certain disordered systems in statistical physics, where the microscopic contribution of each particle is insensitive to remote perturbations of the system. Mathematically, such a lack of long-range interactions can be formalized into a continuity statement with respect to the topology of local weak convergence of graphs. Among other consequences, continuous invariants are guaranteed to admit a deterministic limit along most of the classical sequences of sparse random graphs, and to be efficiently approximable via local distributed algorithms, regardless of the size of the global structure. In this thesis, we focus on four graph invariants that play an important role in theory and applications : the empirical spectral distribution, the kernel dimension of the adjacency matrix, the matching number, and the generating polynomial of certain classes of spanning subgraphs. Each of these notions is shown to admit a unique locally self-consistent extension to local weak limits of finite graphs, and this extension is proven to be continuous. When specialized to the classical models of sparse random graphs, the limiting system of local self-consistency equations simplifies into a single distributional equation, which we solve explicitly. This leads to new asymptotic formulae and to the simplification, unification and generalization of various results that were previously relying on model-specific arguments.
Document type :
Theses
Mathematics [math]. Université Pierre et Marie Curie - Paris VI; Ecole Normale Supérieure de Paris - ENS Paris, 2011. English


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Submitted on : Sunday, October 30, 2011 - 10:06:46 PM
Last modification on : Wednesday, November 2, 2011 - 10:39:55 AM

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Justin Salez. Some implications of local weak convergence for sparse random graphs. Mathematics [math]. Université Pierre et Marie Curie - Paris VI; Ecole Normale Supérieure de Paris - ENS Paris, 2011. English. <tel-00637130>

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