Tight Bounds for Influence in Diffusion Networks and Application to Bond Percolation and Epidemiology

TL;DR

This paper establishes theoretical bounds on the long-term influence of nodes in diffusion networks, linking influence to spectral properties, and applies these results to epidemiology and percolation, providing insights into critical thresholds and network behavior.

Contribution

It introduces spectral radius-based bounds for influence in diffusion models and applies them to epidemiology and percolation, revealing sub-critical regimes and critical thresholds.

Findings

Influence bounds relate to spectral radius of a specific matrix.

Sub-critical influence behavior is O(√n) in general networks.

Derived bounds for critical values in percolation and epidemiology.

Abstract

In this paper, we derive theoretical bounds for the long-term influence of a node in an Independent Cascade Model (ICM). We relate these bounds to the spectral radius of a particular matrix and show that the behavior is sub-critical when this spectral radius is lower than $1$. More specifically, we point out that, in general networks, the sub-critical regime behaves in $O(\sqrt{n})$ where $n$ is the size of the network, and that this upper bound is met for star-shaped networks. We apply our results to epidemiology and percolation on arbitrary networks, and derive a bound for the critical value beyond which a giant connected component arises. Finally, we show empirically the tightness of our bounds for a large family of networks.