Understanding the temporal pattern of spreading in heterogeneous networks: Theory of the mean infection time

TL;DR

This paper introduces a theoretical framework to predict the mean infection time in heterogeneous networks, revealing a slow algebraic spreading phase driven by small-degree nodes, which improves understanding of epidemic dynamics.

Contribution

It develops a method to evaluate the mean infection time using boundary area scaling, addressing fluctuations in cluster size and characterizing the intermediate spreading phase.

Findings

Mean infection time is more reliable than cluster size in heterogeneous networks.

Identifies a slow algebraic spreading phase in the intermediate stage.

Spreading is driven by small-degree nodes remaining susceptible.

Abstract

For a reliable prediction of an epidemic or information spreading pattern in complex systems, well-defined measures are essential. In the susceptible-infected model on heterogeneous networks, the cluster of infected nodes in the intermediate-time regime exhibits too large fluctuation in size to use its mean size as a representative value. The cluster size follows quite a broad distribution, which is shown to be derived from the variation of the cluster size with the time when a hub node was first infected. On the contrary, the distribution of the time taken to infect a given number of nodes is well concentrated at its mean, suggesting the mean infection time is a better measure. We show that the mean infection time can be evaluated by using the scaling behaviors of the boundary area of the infected cluster and use it to find a non-exponential but algebraic spreading phase in the intermediate stage on strongly heterogeneous networks. Such slow spreading originates in only small-degree nodes left susceptible, while most hub nodes are already infected in the early exponential-spreading stage. Our results offer a way to detour around large statistical fluctuations and quantify reliably the temporal pattern of spread under structural heterogeneity.