Inhomogeneous percolation models for spreading phenomena in random graphs

Abstract

Percolation theory has been used a great deal in the study of structural properties ofcomplex networks such as the robustness, with remarkable results. Nevertheless, a purelytopological description is not sufficient for a correct characterization of network behaviourin relation to physical flows and spreading phenomena taking place on them. Thefunctionality of real networks also depends on the ability of the nodes and the edges tobear and handle loads of flows, energy, information and other physical quantities. Wepropose to study these properties, introducing a process of inhomogeneous percolation, inwhich both the nodes and the edges spread the flows out with a given probability.The generating functions approach is exploited in order to get a generalization of theMolloy–Reed criterion for inhomogeneous joint site–bond percolation in correlated randomgraphs. A series of simple assumptions allows the analysis of more realistic situations, forwhich a number of new results are presented. In particular, for the site percolationwith inhomogeneous edge transmission, we obtain the explicit expressions for thepercolation threshold for many interesting cases, which are analysed by means of simpleexamples and numerical simulations. Some possible applications are debated.