A functional central limit theorem for SI processes on configuration model graphs

KhudaBukhsh, Wasiur R.; Woroszylo, Casper; Rempała, Grzegorz A.; Koeppl, Heinz

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. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.

Citation

KhudaBukhsh, W. R., Woroszylo, C., Rempała, G. A., & Koeppl, H. (2022). A functional central limit theorem for SI processes on configuration model graphs. Advances in Applied Probability, 54(3), 880-912. https://doi.org/10.1017/apr.2022.52

• Journal Article Type: Article
• Acceptance Date: Sep 25, 2021
• Online Publication Date: Sep 6, 2022
• Publication Date: Sep 6, 2022
• Deposit Date: Sep 9, 2022
• Publicly Available Date: Mar 7, 2023
• Journal: Advances in Applied Probability
• Print ISSN: 0001-8678
• Electronic ISSN: 1475-6064
• Publisher: Applied Probability Trust
• Peer Reviewed: Peer Reviewed
• Volume: 54
• Issue: 3
• Pages: 880-912
• DOI: https://doi.org/10.1017/apr.2022.52
• Keywords: Applied Mathematics; Statistics and Probability
• Public URL: https://nottingham-repository.worktribe.com/output/10919587
• Publisher URL: https://www.cambridge.org/core/journals/advances-in-applied-probability/article/abs/functional-central-limit-theorem-for-si-processes-on-configuration-model-graphs/8ECE68C7706243E769C7B4BF565C8F3E
• Additional Information: Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust