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

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

We study the lower tail large deviation problem for subgraph counts in a random graph. Let $X_H$ denote the number of copies of $H$ in an Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed $0 < \delta < 1$. 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 $p \ge n^{-\alpha_H}$ (and conjecturally for a larger range of $p$). 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 $H$, and $0 < \delta < \delta_H$ for some $\delta_H > 0$, as $p \to 0$ 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 $H$ and $\delta$ close to 1.Comment: 15 pages, 5 figures, 1 tabl

Ground States for Exponential Random Graphs

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

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