8,292 research outputs found

    Critical percolation on random regular graphs

    Get PDF
    We show that for all d∈{3,…,n−1}d\in \{3,\ldots,n-1\} the size of the largest component of a random dd-regular graph on nn vertices around the percolation threshold p=1/(d−1)p=1/(d-1) is Θ(n2/3)\Theta(n^{2/3}), with high probability. This extends known results for fixed d≥3d\geq 3 and for d=n−1d=n-1, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random dd-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method.Comment: 10 page

    Explicit isoperimetric constants and phase transitions in the random-cluster model

    Full text link
    The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter q≥1q\geq 1. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w (q) for qq large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs

    Interlacement percolation on transient weighted graphs

    Full text link
    In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u_* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the product GxZ (where we endow Z with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value u_*.Comment: 25 pages, 2 figures, accepted for publication in the Elect. Journal of Pro

    Generalized Threshold-Based Epidemics in Random Graphs: the Power of Extreme Values

    Get PDF
    Bootstrap percolation is a well-known activation process in a graph, in which a node becomes active when it has at least rr active neighbors. Such process, originally studied on regular structures, has been recently investigated also in the context of random graphs, where it can serve as a simple model for a wide variety of cascades, such as the spreading of ideas, trends, viral contents, etc. over large social networks. In particular, it has been shown that in G(n,p)G(n,p) the final active set can exhibit a phase transition for a sub-linear number of seeds. In this paper, we propose a unique framework to study similar sub-linear phase transitions for a much broader class of graph models and epidemic processes. Specifically, we consider i) a generalized version of bootstrap percolation in G(n,p)G(n,p) with random activation thresholds and random node-to-node influences; ii) different random graph models, including graphs with given degree sequence and graphs with community structure (block model). The common thread of our work is to show the surprising sensitivity of the critical seed set size to extreme values of distributions, which makes some systems dramatically vulnerable to large-scale outbreaks. We validate our results running simulation on both synthetic and real graphs
    • …
    corecore