Exact solutions for social and biological contagion models on mixed directed and undirected, degree-correlated random networks

TL;DR

This paper derives exact analytical expressions for the likelihood, probability, and expected size of contagion outbreaks in complex networks with mixed directed and undirected edges, considering degree correlations.

Contribution

It extends previous models by providing analytical solutions for contagion processes on mixed directed and undirected networks with degree correlations.

Findings

Derived formulas for contagion spread probabilities and sizes.

Numerical validation on specific network classes.

Extended theoretical framework beyond undirected networks.

Abstract

We derive analytic expressions for the possibility, probability, and expected size of global spreading events starting from a single infected seed for a broad collection of contagion processes acting on random networks with both directed and undirected edges and arbitrary degree-degree correlations. Our work extends previous theoretical developments for the undirected case, and we provide numerical support for our findings by investigating an example class of networks for which we are able to obtain closed-form expressions.