2,865 research outputs found

    Sampling in Potts Model on Sparse Random Graphs

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    We study the problem of sampling almost uniform proper q-colorings in sparse Erdos-Renyi random graphs G(n,d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [Dyer et al., RANDOM STRUCT ALGOR, 2006]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q>3d+4, improving the current best bound q>5.5d [Efthymiou, SODA, 2014]. Our sampling algorithm works for more generalized models and broader family of sparse graphs. It is an efficient sampler (in the same sense of FPAUS) for anti-ferromagnetic Potts model with activity 03(1-b)d+4. We further identify a family of sparse graphs to which all these results can be extended. This family of graphs is characterized by the notion of contraction function, which is a new measure of the average degree in graphs

    Factor models on locally tree-like graphs

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    We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree TT, and study the existence of the free energy density ϕ\phi, the limit of the log-partition function divided by the number of vertices nn as nn tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ\phi subject to uniqueness of a relevant Gibbs measure for the factor model on TT. By way of example we compute ϕ\phi for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ\phi. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on TT. In the special case that TT has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Statistical Mechanics of Community Detection

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    Starting from a general \textit{ansatz}, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the \textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the modularity QQ as defined by Newman and Girvan \cite{Girvan03} as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the \textit{ansatz} may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure

    Minimizing Unsatisfaction in Colourful Neighbourhoods

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    Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods, given a predetermined number of colours. In the analytical framework of a tree approximation, carried out at both zero and finite temperatures, solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition, which are consistent with those of existing algorithms. The nature of the transition as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional explanatio

    Sidorenko's conjecture, colorings and independent sets

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    Let hom(H,G)\hom(H,G) denote the number of homomorphisms from a graph HH to a graph GG. Sidorenko's conjecture asserts that for any bipartite graph HH, and a graph GG we have hom(H,G)v(G)v(H)(hom(K2,G)v(G)2)e(H),\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)}, where v(H),v(G)v(H),v(G) and e(H),e(G)e(H),e(G) denote the number of vertices and edges of the graph HH and GG, respectively. In this paper we prove Sidorenko's conjecture for certain special graphs GG: for the complete graph KqK_q on qq vertices, for a K2K_2 with a loop added at one of the end vertices, and for a path on 33 vertices with a loop added at each vertex. These cases correspond to counting colorings, independent sets and Widom-Rowlinson colorings of a graph HH. For instance, for a bipartite graph HH the number of qq-colorings ch(H,q)\textrm{ch}(H,q) satisfies ch(H,q)qv(H)(q1q)e(H).\textrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}. In fact, we will prove that in the last two cases (independent sets and Widom-Rowlinson colorings) the graph HH does not need to be bipartite. In all cases, we first prove a certain correlation inequality which implies Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande
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