Asymptotic Structure of Constrained Exponential Random Graph Models

Zhu, Lingjiong

doi:10.1007/s10955-017-1733-y

Cited by 6 publications

(7 citation statements)

References 23 publications

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“…Most papers focus on dense graphs, but there are some interesting advances for sparse graphs as well. Closely related to the canonical ensemble are the exponential random graph model (Bhamidi et al [3], Chatterjee and Diaconis [9]) and the constrained exponential random model (Aristoff and Zhu [1], Kenyon and Yin [19], Yin [30], Zhu [32]).…”

Section: Relevant Literaturementioning

confidence: 99%

Ensemble equivalence for dense graphs

Hollander¹,

Mandjes²,

Roccaverde³

et al. 2018

Electron. J. Probab.

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In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices n. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as n grows large, where two ensembles are said to be equivalent in the dense regime if this relative entropy divided by n 2 tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are frustrated. Examples are provided for three different choices of constraints.

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Section: Relevant Literaturementioning

confidence: 99%

Ensemble equivalence for dense graphs

Hollander¹,

Mandjes²,

Roccaverde³

et al. 2018

Electron. J. Probab.

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“…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…”

Section: The Canonical Ensemblementioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Reciprocity in directed networks

Yin

Zhu

2016

Physica A: Statistical Mechanics and its Applications

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Reciprocity is an important characteristic of directed networks and has been widely used in the modeling of World Wide Web, email, social, and other complex networks. In this paper, we take a statistical physics point of view and study the limiting entropy and free energy densities from the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble whose sufficient statistics are given by edge and reciprocal densities. The sparse case is also studied for the grand canonical ensemble. Extensions to more general reciprocal models including reciprocal triangle and star densities will likewise be discussed.

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“…See e.g. Chatterjee and Varadhan [13], Chatterjee and Diaconis [12], Radin and Yin [29], Lubetzky and Zhao [24], Radin and Sadun [27,28], Radin et al [26], Kenyon et al [19], Yin [35], Yin et al [36], Kenyon and Yin [20], Aristoff and Zhu [2,3], and Zhu [37]. Most of these papers utilize the theory of graph limits as developed by Lovász and coauthors (V. T. Sós, B. Szegedy, C. Borgs, J. Chayes, K. Vesztergombi, etc.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for sparse exponential random graph models

Yin¹,

Zhu²

2017

Braz. J. Probab. Stat.

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Abstract. We study the asymptotics for sparse exponential random graph models where the parameters may depend on the number of vertices of the graph. We obtain exact estimates for the mean and variance of the limiting probability distribution and the limiting log partition function of the edge-(single)-star model. They are in sharp contrast to the corresponding asymptotics in dense exponential random graph models. Similar analysis is done for directed sparse exponential random graph models parametrized by edges and multiple outward stars.

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Asymptotic Structure of Constrained Exponential Random Graph Models

Cited by 6 publications

References 23 publications

Ensemble equivalence for dense graphs

Ensemble equivalence for dense graphs

Reciprocity in directed networks

Asymptotics for sparse exponential random graph models

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