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

A Stochastic Model of Active Cyber Defense Dynamics

The concept of active cyber defense has been proposed for years. However, there are no mathematical models for characterizing the effectiveness of active cyber defense. In this paper, we fill the void by proposing a novel Markov process model that is native to the interaction between cyber attack and active cyber defense. Unfortunately, the native Markov process model cannot be tackled by the techniques we are aware of. We therefore simplify, via mean-field approximation, the Markov process model as a Dynamic System model that is amenable to analysis. This allows us to derive a set of valuable analytical results that characterize the effectiveness of four types of active cyber defense dynamics. Simulations show that the analytical results are inherent to the native Markov process model, and therefore justify the validity of the Dynamic System model. We also discuss the side-effect of the mean-field approximation and its implications

The phase transition in inhomogeneous random graphs

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that under one very weak assumption (that the expected number of edges is `what it should be'), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is `stable': for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random Structures and Algorithm

Discontinuous Percolation Transitions in Epidemic Processes, Surface Depinning in Random Media and Hamiltonian Random Graphs

Discontinuous percolation transitions and the associated tricritical points are manifest in a wide range of both equilibrium and non-equilibrium cooperative phenomena. To demonstrate this, we present and relate the continuous and first order behaviors in two different classes of models: The first are generalized epidemic processes (GEP) that describe in their spatially embedded version - either on or off a regular lattice - compact or fractal cluster growth in random media at zero temperature. A random graph version of GEP is mapped onto a model previously proposed for complex social contagion. We compute detailed phase diagrams and compare our numerical results at the tricritical point in d = 3 with field theory predictions of Janssen et al. [Phys. Rev. E 70, 026114 (2004)]. The second class consists of exponential ("Hamiltonian", or formally equilibrium) random graph models and includes the Strauss and the 2-star model, where 'chemical potentials' control the densities of links, triangles or 2-stars. When the chemical potentials in either graph model are O(logN), the percolation transition can coincide with a first order phase transition in the density of links, making the former also discontinuous. Hysteresis loops can then be of mixed order, with second order behavior for decreasing link fugacity, and a jump (first order) when it increases

Principal Patterns on Graphs: Discovering Coherent Structures in Datasets

Graphs are now ubiquitous in almost every field of research. Recently, new research areas devoted to the analysis of graphs and data associated to their vertices have emerged. Focusing on dynamical processes, we propose a fast, robust and scalable framework for retrieving and analyzing recurring patterns of activity on graphs. Our method relies on a novel type of multilayer graph that encodes the spreading or propagation of events between successive time steps. We demonstrate the versatility of our method by applying it on three different real-world examples. Firstly, we study how rumor spreads on a social network. Secondly, we reveal congestion patterns of pedestrians in a train station. Finally, we show how patterns of audio playlists can be used in a recommender system. In each example, relevant information previously hidden in the data is extracted in a very efficient manner, emphasizing the scalability of our method. With a parallel implementation scaling linearly with the size of the dataset, our framework easily handles millions of nodes on a single commodity server