2,698 research outputs found

    Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality

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    Conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only. Since many substantively important networks are weighted, this limitation is especially problematic. We extend the existing exponential framework by proposing a generic common distribution for the edge weights. Minimal assumptions are placed on the distribution, that is, it is non-degenerate and supported on the unit interval. By doing so, we recognize the essential properties associated with near-degeneracy and universality in edge-weighted exponential random graphs.Comment: 15 pages, 4 figures. This article extends arXiv:1607.04084, which derives general formulas for the normalization constant and characterizes phase transitions in exponential random graphs with uniformly distributed edge weights. The present article places minimal assumptions on the edge-weight distribution, thereby recognizing essential properties associated with near-degeneracy and universalit

    On the lower tail variational problem for random graphs

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    We study the lower tail large deviation problem for subgraph counts in a random graph. Let XHX_H denote the number of copies of HH in an Erd\H{o}s-R\'enyi random graph G(n,p)\mathcal{G}(n,p). We are interested in estimating the lower tail probability P(XH≤(1−δ)EXH)\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H) for fixed 0<δ<10 < \delta < 1. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p≥n−αHp \ge n^{-\alpha_H} (and conjecturally for a larger range of pp). We study this variational problem and provide a partial characterization of the so-called "replica symmetric" phase. Informally, our main result says that for every HH, and 0<δ<δH0 < \delta < \delta_H for some δH>0\delta_H > 0, as p→0p \to 0 slowly, the main contribution to the lower tail probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite HH and δ\delta close to 1.Comment: 15 pages, 5 figures, 1 tabl

    Ground States for Exponential Random Graphs

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    We propose a perturbative method to estimate the normalization constant in exponential random graph models as the weighting parameters approach infinity. As an application, we give evidence of discontinuity in natural parametrization along the critical directions of the edge-triangle model.Comment: 12 pages, 3 figures, 1 tabl

    A detailed investigation into near degenerate exponential random graphs

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    The exponential family of random graphs has been a topic of continued research interest. Despite the relative simplicity, these models capture a variety of interesting features displayed by large-scale networks and allow us to better understand how phases transition between one another as tuning parameters vary. As the parameters cross certain lines, the model asymptotically transitions from a very sparse graph to a very dense graph, completely skipping all intermediate structures. We delve deeper into this near degenerate tendency and give an explicit characterization of the asymptotic graph structure as a function of the parameters.Comment: 15 pages, 3 figures, 2 table
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