Multivariate Generating Functions for Information Spread on Multi-Type Random Graphs

TL;DR

This paper develops a mathematical framework using multivariate generating functions and multi-type branching processes to analyze how information and epidemics spread across complex multi-type directed random graphs with diverse community structures.

Contribution

It introduces a novel approach combining multivariate generating functions and branching processes to model multi-type random graphs with general degree distributions.

Findings

Derived an equation for the size of large out-components in multi-type graphs.

Analyzed epidemic spread dynamics across different community types.

Validated theoretical results with population-based simulations.

Abstract

We study the spread of information on multi-type directed random graphs. In such graphs the vertices are partitioned into distinct types (communities) that have different transmission rates between themselves and with other types. We construct multivariate generating functions and use multi-type branching processes to derive an equation for the size of the large out-components in multi-type random graphs with a general class of degree distributions. We use our methods to analyse the spread of epidemics and verify the results with population based simulations