Inhomogeneous percolation models for spreading phenomena in random graphs

Percolation theory has been used a great deal 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 network behaviour in relation to 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 to bear and handle 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 the flows out with a given probability. The 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 for the percolation threshold for many interesting cases, which are analysed by means of simple examples and numerical simulations. Some possible applications are debated.