Multipodal Structure and Phase Transitions in Large Constrained Graphs

Abstract

We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of k-star subgraphs, k ≥ 2 fixed. We prove that under such constraints graphs are "multipodal": asymptotically in the number of vertices there is a partition of the vertices into M < ∞ subsets V 1 , V 2 , . . . , V M , and a set of well-defined probabilities g ij of an edge between any v i ∈ V i and v j ∈ V j . For 2 ≤ k ≤ 30 we determine the phase space: the combinations of edge and k-star densities achievable asymptotically. For these models there are special points on the boundary of the phase space with nonunique asymptotic (graphon) structure; for the 2-star model we prove that the nonuniqueness extends to entropy maximizers in the interior of the phase space.

Reciprocity in directed networks

“…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. The optimal graphon g is multipodal and phase transitions are expected.…”

“…Let us analyze the monotonicity of the limiting entropy density. On one hand, 17) which implies that ∂λ ∂ ≥ 0 if and only if − r ≤ 1 + r − 2 , which is equivalent to 1 + 2r ≥ 3 . On the other hand, 18) which implies that ∂λ ∂r ≥ 0 if and only if r(1 + r − 2 ) ≤ ( − r) 2 , which is equivalent to r ≤ 2 , i.e., λ( , r) is increasing in r below the Erdős-Rényi curve and decreasing in r above the Erdős-Rényi curve.…”

“…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].…”

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].…”

“…Before we proceed, let us mention an alternative to exponential random graph models that was introduced by Radin and Sadun [16], where instead of using parameters to control subgraph counts, the subgraph densities are controlled directly; see also Radin et al [17], Radin and Sadun [18] and Kenyon et al [9]. For example, we can fix the edge density and the density of a given simple finite graph H and study the entropy…”