Stability of Spreading Processes over Time-Varying Large-Scale Networks

TL;DR

This paper introduces a flexible model for time-varying networks that accurately captures arbitrary inter-switching times and analyzes the stability of spreading processes, providing more precise conditions than traditional Markovian models.

Contribution

We develop an extended family of network processes with arbitrary inter-switching time distributions and derive scalable stability conditions for spreading processes on these networks.

Findings

Stability conditions are derived using eigenvalues of matrices with size linear in the number of nodes.

Heuristics based on static network aggregation improve epidemic threshold approximation as network size increases.

Numerical simulations validate the theoretical stability conditions and insights.

Abstract

In this paper, we analyze the dynamics of spreading processes taking place over time-varying networks. A common approach to model time-varying networks is via Markovian random graph processes. This modeling approach presents the following limitation: Markovian random graphs can only replicate switching patterns with exponential inter-switching times, while in real applications these times are usually far from exponential. In this paper, we introduce a flexible and tractable extended family of processes able to replicate, with arbitrary accuracy, any distribution of inter-switching times. We then study the stability of spreading processes in this extended family. We first show that a direct analysis based on It\^o's formula provides stability conditions in terms of the eigenvalues of a matrix whose size grows exponentially with the number of edges. To overcome this limitation, we derive alternative stability conditions involving the eigenvalues of a matrix whose size grows linearly with the number of nodes. Based on our results, we also show that heuristics based on aggregated static networks approximate the epidemic threshold more accurately as the number of nodes grows, or the temporal volatility of the random graph process is reduced. Finally, we illustrate our findings via numerical simulations.