Inhomogeneous percolation models for spreading phenomena in random graphs

TL;DR

This paper extends percolation theory to inhomogeneous models on complex networks, incorporating flow-bearing capacities of nodes and edges, and derives new thresholds and criteria relevant for real-world spreading phenomena.

Contribution

It introduces an inhomogeneous percolation framework for networks, generalizes the Molloy-Reed Criterion, and provides explicit thresholds for realistic scenarios.

Findings

Derived explicit percolation thresholds for inhomogeneous edge transmission.

Generalized Molloy-Reed Criterion for correlated random graphs.

Analyzed realistic cases with numerical simulations.

Abstract

Percolation theory has been largely used in the study of structural properties of complex networks such as the robustness, with remarkable results. Nevertheless, a purely topological description is not sufficient for a correct characterization of networks behaviour in relation with physical flows and spreading phenomena taking place on them. The functionality of real networks also depends on the ability of the nodes and the edges in bearing and handling loads of flows, energy, information and other physical quantities. We propose to study these properties introducing a process of inhomogeneous percolation, in which both the nodes and the edges spread out the flows with a given probability. Generating functions approach is exploited in order to get a generalization of the Molloy-Reed Criterion for inhomogeneous joint site bond percolation in correlated random graphs. A series of simple assumptions allows the analysis of more realistic situations, for which a number of new results are presented. In particular, for the site percolation with inhomogeneous edge transmission, we obtain the explicit expressions of the percolation threshold for many interesting cases, that are analyzed by means of simple examples and numerical simulations. Some possible applications are debated.