[Paper Review] Functional Central Limit Theorem For Susceptible-Infected Process On Configuration Model Graphs

Wasiur R. KhudaBukhsh, Casper Woroszylo+2 more|arXiv (Cornell University)|Mar 18, 2017

Complex Network Analysis Techniques45 references8 citations

TL;DR

This paper establishes a Functional Central Limit Theorem (FCLT) for the susceptible-infected (SI) epidemic process on configuration model random graphs. As the number of vertices grows, the scaled counts of susceptible, infected, SI-edges, and SS-edges converge weakly to a continuous Gaussian vector semimartingale, providing a diffusion approximation for epidemic dynamics on complex networks.

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.

Motivation & Objective

• To derive a functional central limit theorem (FCLT) for the SI epidemic process on configuration model (CM) random graphs.
• To model the joint dynamics of susceptible/infectious individuals and edge types (SI, SS) in a finite-time stochastic process.
• To establish weak convergence of scaled fluctuations around the functional law of large numbers (FLLN) limit to a Gaussian semimartingale.
• To provide a rigorous scaling limit for stochastic epidemic dynamics on sparse, heterogeneous networks.
• To support applications in percolation theory and computer virus modeling via diffusion approximations.

Proposed method

• The process is described via aggregated state variables: $X_S(t)$ (susceptible count), $X_{SI}(t)$ (SI-edge count), and $X_{SS}(t)$ (SS-edge count, double-counted).
• The fluctuation process $Y(t) = \sqrt{n}(n^{-1}X(t) - x(t))$ is analyzed, where $x(t)$ is the FLLN limit solution to a system of ODEs.
• Rebolledo's theorem for locally square-integrable martingales is applied, requiring convergence of predictable and optional quadratic variation processes.
• The proof relies on verifying three conditions: convergence of predictable quadratic variation $\langle M_n \rangle \to V$, optional quadratic variation $[M_n] \to V$, and negligible jump contributions $\langle M_n^{(\epsilon)} \rangle \to 0$.
• The limiting process is a 3-dimensional continuous Gaussian vector semimartingale in the Skorohod space $D^{(3)}$ of cadlag functions on $[0,T]$.
• Technical assumptions on the degree distribution and initial conditions ensure the convergence under general settings.

Experimental results

Research questions

• RQ1 Does the joint process of susceptible individuals and SI/SS edges on a configuration model graph satisfy a functional central limit theorem as the network size grows?
• RQ2 Can the fluctuations of the SI epidemic process around its fluid limit be approximated by a diffusion process?
• RQ3 What are the necessary and sufficient conditions on the degree distribution and initial state for the FCLT to hold?
• RQ4 How do the edge-type counts (SI and SS) contribute to the diffusion approximation of epidemic spread?
• RQ5 Can the limiting Gaussian process be used to model real-world phenomena such as computer virus propagation or percolation?

Key findings

• The scaled process $\sqrt{n}(n^{-1}X(t) - x(t))$ converges weakly to a continuous Gaussian vector semimartingale in the Skorohod topology on $[0,T]$.
• The limiting process has a predictable quadratic variation matrix $V(t)$ that is continuous and deterministic, with positive semidefinite increments.
• The convergence holds under technical conditions on the degree distribution, including finite second moments and regularity of initial configurations.
• The FCLT applies to general degree distributions, including power-law and Poisson, as validated by simulation results.
• The limiting Gaussian process captures the joint fluctuations of compartment sizes and edge types, enabling statistical inference and simulation of epidemic dynamics.
• The result provides a rigorous foundation for diffusion approximations in network epidemiology and percolation theory.