Phase transitions in exponential random graphs

Abstract

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.Comment: Published in at http://dx.doi.org/10.1214/12-AAP907 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Detailed Investigation into Near Degenerate Exponential Random Graphs

“…The double asymptotic framework of [11] was later extended in [26], and the scenario in which the parameters diverge along general straight lines β 1 = aβ 2 , where a is a constant and β 2 → ∞, was considered. Consistent with the near degeneracy predictions in [3,11,24], asymptotically for a ≤ −1, a typical graph sampled from the "attractive" 2-parameter exponential model is sparse, while for a > −1, a typical graph is nearly complete. Although much effort has been focused on 2-parameter models, k-parameter models have also been examined.…”

“…In further works (see for example, Radin and Yin [24], Aristoff and Zhu [3]), this singular behavior was discovered universally in generic 2-parameter models where H 1 is an edge and H 2 is any finite simple graph, and the transition curve β 2 = q(β 1 ) asymptotically approaches the straight line β 2 = −β 1 as the parameters diverge. The double asymptotic framework of [11] was later extended in [26], and the scenario in which the parameters diverge along general straight lines β 1 = aβ 2 , where a is a constant and β 2 → ∞, was considered.…”

Asymptotic Structure of Constrained Exponential Random Graph Models

“…¿From the properties of β 1 x + β 2 x p − I(x) studied in [15], [2], [3], for β 2 ≤ (p−1) p , as β 1 increases from −∞ to q −1 (β 2 ), the maximizer of β 1 x+β 2 x p −I(x) increases from 0 to x 1 , and as β 1 increases from q −1 (β 2 ) to ∞, the maximizer of β 1 x + β 2 x p − I(x) increases from x 2 to 1, where 0 < x 1 < x 2 < 1 are the two maximizers of β 1 x + β 2 x p − I(x) for β 1 = q −1 (β 2 ). Hence, we proved that the optimizing graphon in the canonical model for β 2 ≥ 0 is uniform if ( , 2β 2 ) is outside of the U -shaped region as in Proposition 5.…”

“…The emphasis has been made on the limiting free energy and entropy, phase transitions and asymptotic structures, see e.g. Chatterjee and Diaconis [5], Radin and Yin [15], Radin and Sadun [16], Radin et al [17], Radin and Sadun [18], Kenyon et al [9], Yin [22], Yin et al [23], Aristoff and Zhu [2], Aristoff and Zhu [3]. In this paper, we are interested to study the constrained exponential random graph models introduced in Kenyon and Yin [10].…”

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.…”