Asymptotic quantization of exponential random graphs

Abstract

We describe the asymptotic properties of the edge-triangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. As a result, we provide a complete description of all asymptotic extremal behaviors of the model.Comment: 38 pages, 7 figure

A Detailed Investigation into Near Degenerate Exponential Random Graphs

“…We would like to know if these asymptotic results could be generalized. By [26], taking β 1 = aβ 2 and β 2 → ∞, u * → 0 when a ≤ −1 and u * → 1 when a > −1. Equivalently, for (β 1 , β 2 ) sufficiently far away from the origin, u * → 0 when β 1 ≤ −β 2 and u * → 1 when β 1 > −β 2 .…”

“…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. 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.…”

Asymptotic Structure of Constrained Exponential Random Graph Models

“…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].…”

Reciprocity in directed networks

“…See e.g. Aristoff and Zhu [2,3], Chatterjee and Dembo [5], Chatterjee and Diaconis [6], Kenyon et al [17], Kenyon and Yin [18], Lubetzky and Zhao [22,23], Radin and Sadun [27,28], Radin et al [26], Radin and Yin [29], Yin [35], Yin et al [36], and Zhu [38]. It may be worth pointing out that most of these papers utilize the theory of graph limits as developed by Lovász and coworkers [20,21].…”