Critical Phenomena in Exponential Random Graphs

Abstract

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. By adjusting the specific values of these subgraph densities, one can analyze the influence of various local features on the global structure of the network. Loosely put, a phase transition occurs when a singularity arises in the limiting free energy density, as it is the generating function for the limiting expectations of all thermodynamic observables. We derive the full phase diagram for a large family of 3-parameter exponential random graph models with attraction and show that they all consist of a first order surface phase transition bordered by a second order critical curve.

A Detailed Investigation into Near Degenerate Exponential Random Graphs

“…Although much effort has been focused on 2-parameter models, k-parameter models have also been examined. As shown in [25], near degeneracy and universality are expected not only in generic 2-parameter models but also in generic k-parameter models. Asymptotically, a typical graph drawn from the "attractive" k-parameter exponential model where β 2 , .…”

Reciprocity in directed networks

“…Many people have delved into this area. A particularly significant discovery was made by Chatterjee and Diaconis [6], who showed that the supremum in (5.16) is always attained and a random graph drawn from the model must lie close to the maximizing set with probability vanishing in n. When β 3 , β 4 ≥ 0, Yin [35] further showed that the 3-parameter space would consist of a single phase with first-order phase transition(s) across one (or more) surfaces, where all the first derivatives of χ exhibit (jump) discontinuities, and second-order phase transition(s) along one (or more) critical curves, where all the second derivatives of χ diverge. The second special situation is when β 3 = 0, Following similar arguments as in Kenyon et al [17], we conclude that d(x) can take only finitely many values.…”

An introduction to large deviations for random graphs

“…For a few mathematical results preceding [19], see [4,18]. For a nonexhaustive list of subsequent developments, see [2,3,30,31,[41][42][43][44][45][46]51]. The discussion in this section will be limited to a basic result from [19] and one easy example.…”