Logistic growth on networks: exact solutions for the SI model

TL;DR

This paper presents an exact analytical solution for the SI model of information spreading on networks, using a novel approach inspired by quantum many-body systems, revealing detailed insights beyond mean-field approximations.

Contribution

Introduces a new method to solve the SI model on networks exactly by organizing node states through subgraph contributions and symmetry relations.

Findings

Exact solutions match Monte-Carlo simulations

Results differ significantly from mean-field approximations

Method applicable to various network topologies

Abstract

The SI model is the most basic of all compartmental models used to describe the spreading of information through a population. Despite its apparent simplicity, the analytic solution of this model on networks is still lacking. We address this problem here, using a novel formulation inspired by the mathematical treatment of many-body quantum systems. This allows us to organize the time-dependent expectation values for the state of individual nodes in terms of contributions from subgraphs of the network. We compute these contributions systematically and find a set of symmetry relations among subgraphs of differing topologies. We use our novel approach to compute the spreading of information on three different sample networks. The exact solution, which matches with Monte-Carlo simulations, visibly departs from the mean-field results.