Inhomogeneous percolation models for spreading phenomena in random graphs

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.Comment: 28 pages, 11 figure

Consensus formation on coevolving networks: groups' formation and structure

We study the effect of adaptivity on a social model of opinion dynamics and consensus formation. We analyze how the adaptivity of the network of contacts between agents to the underlying social dynamics affects the size and topological properties of groups and the convergence time to the stable final state. We find that, while on static networks these properties are determined by percolation phenomena, on adaptive networks the rewiring process leads to different behaviors: Adaptive rewiring fosters group formation by enhancing communication between agents of similar opinion, though it also makes possible the division of clusters. We show how the convergence time is determined by the characteristic time of link rearrangement. We finally investigate how the adaptivity yields nontrivial correlations between the internal topology and the size of the groups of agreeing agents.Comment: 10 pages, 3 figures,to appear in a special proceedings issue of J. Phys. A covering the "Complex Networks: from Biology to Information Technology" conference (Pula, Italy, 2007

Explosive percolation in graphs

Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the same probability. However, alternative rules for the occupation of sites/bonds might affect the order of the transition. A recent set of rules proposed by Achlioptas et al. [Science 323, 1453 (2009)], characterized by competitive link addition, was claimed to lead to a discontinuous connectedness transition, named "explosive percolation". In this work we survey a numerical study of the explosive percolation transition on various types of graphs, from lattices to scale-free networks, and show the consistency of these results with recent analytical work showing that the transition is actually continuous.Comment: 10 pages, 7 figures, 1 table. Contribution to the Proceedings of STATPHYS-Kolkata VII, November 26-30, 201

Indirect influence in social networks as an induced percolation phenomenon

Significance Increasing empirical evidence in diverse social and ecological systems has shown that indirect interactions play a pivotal role in shaping systems’ dynamical behavior. Our empirical study on collaboration networks of scientists further reveals that an indirect effect can dominate over direct influence in behavioral spreading. However, almost all models in existence focus on direct interactions, and the general impact of indirect interactions has not been studied. We propose a new percolation process, termed induced percolation, to characterize indirect interactions and find that indirect interactions raise a plethora of new phenomena, including the wide range of possible phase transitions. Such an indirect mechanism leads to very different spreading outcomes from that of direct influences

Switch between critical percolation modes in city traffic dynamics

Percolation transition is widely observed in networks ranging from biology to engineering. While much attention has been paid to network topologies, studies rarely focus on critical percolation phenomena driven by network dynamics. Using extensive real data, we study the critical percolation properties in city traffic dynamics. Our results suggest that two modes of different critical percolation behaviors are switching in the same network topology under different traffic dynamics. One mode of city traffic (during nonrush hours or days off) has similar critical percolation characteristics as small world networks, while the other mode (during rush hours on working days) tends to behave as a 2D lattice. This switching behavior can be understood by the fact that the high-speed urban roads during nonrush hours or days off (that are congested during rush hours) represent effective long-range connections, like in small world networks. Our results might be useful for understanding and improving traffic resilience.Comment: 8 pages, 4 figures, Daqing Li, Ziyou Gao and H. Eugene Stanley are the corresponding authors ([email protected], [email protected], [email protected]