Logistic growth on networks: exact solutions for the SI model

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.

15 pages, 4 figures, accompanied by a software package at https://doi.org/10.6084/m9.figshare.14872182.v4. v2: extended explanation and incorporated supplemental material into the main text. Accepted for publication in Phys.Rev. E

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Cited by in corpus (6)

• Field theories and quantum methods for stochastic reaction-diffusion systems
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• Emergent information dynamics in many-body interconnected systems
• Risk aversion can promote cooperation
• Intermediate time scale in the first product formation time distribution of Michaelis-Menten kinetics with inhibitors
• Universality classes in the time evolution of epidemic outbreaks on complex networks