Functional Central Limit Theorem For Susceptible-Infected Process On Configuration Model Graphs

Abstract

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.

52 pages, 6 figures

References in corpus (2)

• Nonequilibrium phase transition in a spreading process on a timeline
• First passage percolation on random graphs with infinite variance degrees

Cited by in corpus (2)

• Central limit theorems in the configuration model
• Survival Dynamical Systems for the Population-level Analysis of Epidemics