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Register a new userFunctional Central Limit Theorem For Susceptible-Infected Process On Configuration Model Graphs
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We study a stochastic compartmental susceptible-infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval $[0,T],$ for some $ T>0$. In this setting, we split the population of graph nodes into two compartments, namely, $S$ and $I$, denoting the susceptible and infected nodes, respectively. In addition to the sizes of these two compartments, we study counts of $SI$-edges (those connecting a susceptible and an infected node) and $SS$-edges (those connecting two susceptible nodes). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them, a scaling limit of the dynamical process as $n$, the number of nodes in the random graph, grows to infinity. To be precise, we show that these counts, when appropriately scaled, converge weakly to a continuous Gaussian vector martingale process the usual Skorohod space of real 3-dimensional vector-valued cadlag, functions on $[0,T]$ endowed with the Skorohod topology. We assume certain technical requirements for this purpose. We discuss applications of our FCLT in percolation theory (from a non-equilibrium statistical mechanics point of view), and in computer science in the context of spread of computer viruses. We also provide simulation results for some common degree distributions.
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A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and configuration of graphs, where the processes are set on. The explicit formulae for the parameters of asymmetry for the vertices of the limiting graph are given in the case, when, in the pre-limiting graphs, some groups of vertices form knots contracting into a points.
We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of nodes with threshold zero which consists of initially activated nodes, whereas every other node is inactive. Subsequently, in each round, if an inactive node with threshold $theta$ has at least $theta$ of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more nodes become activated. The main result of this paper provides a central limit theorem for the final size of activated nodes. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated nodes has asymptotically Gaussian fluctuations.
We study Random Walks in an i.i.d. Random Environment (RWRE) defined on $b$-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the process is defined on $b$-regular trees, with $ b ge 4$, substantially improving the results of the first author (see Theorem 3 of Collevecchio (2006)).
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The Susceptible-Infected-Susceptible model is a canonical model for emerging disease outbreaks. Such outbreaks are naturally modeled as taking place on networks. A theoretical challenge in network epidemiology is the dynamic correlations coming from that if one node is occupied, or infected (for disease spreading models), then its neighbors are likely to be occupied. By combining two theoretical approaches---the heterogeneous mean-field theory and the effective degree method---we are able to include these correlations in an analytical solution of the SIS model. We derive accurate expressions for the average prevalence (fraction of infected) and epidemic threshold. We also discuss how to generalize the approach to a larger class of stochastic population models.
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