A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon

TL;DR

This paper introduces a flexible random network model with tunable clustering, degree correlation, and degree distribution, and analyzes how these features influence the spread of an epidemic modeled by SIR dynamics.

Contribution

It extends the configuration model to include controllable clustering and degree correlation, providing asymptotic analysis of epidemic behavior on such networks.

Findings

Clustering decreases disease spread.

Degree correlation impacts epidemic size near threshold.

Higher degree correlation increases spread when R* is just above 1.

Abstract

A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (half-edges) are paired to form a network. Clustering is obtained by forming small completely connected subgroups, and positive (negative) degree correlation is obtained by connecting a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR (Susceptible -> Infective -> Recovered) epidemic model is defined on this network. Asymptotic properties of both the network and the epidemic, as the population size tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a reproduction number $R_*$, the probability of a major outbreak and the relative size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. The main findings are that clustering tends to decrease the spread of disease, the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and disease spread broadly increases with degree correlation $\rho$ when $R_*$ is just above its threshold value of one and decreases with $\rho$ when $R_*$ is well above one.