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

Hierarchical models of rigidity percolation

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks, and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.Comment: 20 pages, 8 figures. Figures improved, references added, small modifications. Accepted in Phys. Rev.

Optimal Path and Minimal Spanning Trees in Random Weighted Networks

We review results on the scaling of the optimal path length in random networks with weighted links or nodes. In strong disorder we find that the length of the optimal path increases dramatically compared to the known small world result for the minimum distance. For Erd\H{o}s-R\'enyi (ER) and scale free networks (SF), with parameter $\lambda$ ($\lambda >3$), we find that the small-world nature is destroyed. We also find numerically that for weak disorder the length of the optimal path scales logaritmically with the size of the networks studied. We also review the transition between the strong and weak disorder regimes in the scaling properties of the length of the optimal path for ER and SF networks and for a general distribution of weights, and suggest that for any distribution of weigths, the distribution of optimal path lengths has a universal form which is controlled by the scaling parameter $Z=\ell_{\infty}/A$ where $A$ plays the role of the disorder strength, and $\ell_{\infty}$ is the length of the optimal path in strong disorder. The relation for $A$ is derived analytically and supported by numerical simulations. We then study the minimum spanning tree (MST) and show that it is composed of percolation clusters, which we regard as "super-nodes", connected by a scale-free tree. We furthermore show that the MST can be partitioned into two distinct components. One component the {\it superhighways}, for which the nodes with high centrality dominate, corresponds to the largest cluster at the percolation threshold which is a subset of the MST. In the other component, {\it roads}, low centrality nodes dominate. We demonstrate the significance identifying the superhighways by showing that one can improve significantly the global transport by improving a very small fraction of the network.Comment: review, accepted at IJB

Numerical evaluation of the upper critical dimension of percolation in scale-free networks

We propose a numerical method to evaluate the upper critical dimension $d_c$ of random percolation clusters in Erd\H{o}s-R\'{e}nyi networks and in scale-free networks with degree distribution ${\cal P}(k) \sim k^{-\lambda}$, where $k$ is the degree of a node and $\lambda$ is the broadness of the degree distribution. Our results report the theoretical prediction, $d_c = 2(\lambda - 1)/(\lambda - 3)$ for scale-free networks with $3 < \lambda < 4$ and $d_c = 6$ for Erd\H{o}s-R\'{e}nyi networks and scale-free networks with $\lambda > 4$. When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain $d_c = 6$ for all $\lambda > 2$. Our method also yields a better numerical evaluation of the critical percolation threshold, $p_c$, for scale-free networks. Our results suggest that the finite size effects increases when $\lambda$ approaches 3 from above.Comment: 10 pages, 5 figure

On the freezing of variables in random constraint satisfaction problems

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we study the critical behavior around the freezing transition, which appears in the unfrozen phase as the divergence of the sizes of the rearrangements induced in response to the modification of a variable. The formalism is developed on generic constraint satisfaction problems and applied in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.Comment: 32 pages, 7 figure