On the asymptotics of constrained exponential random graphs

Abstract

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to get concrete answers in the edge-triangle model with fixed density of edges.Comment: 23 pages, 4 figure

Reciprocity in directed networks

“…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 [28] for a similar phenomenon across the Erdős-Rényi curve in the (undirected) edge-triangle model.…”

“…As in Aristoff and Zhu [3], Kenyon and Yin [18] and Zhu [38], we are interested in the asymptotic features of constrained models. The probability measure is given by…”

“…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

“…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]. The directed case was first studied in Aristoff and Zhu [3].…”

“…, k}. We also define t(H, h) = For a more detailed introduction and background about the exponential random graph model, we refer to Section 2 of Kenyon and Yin [10].…”