Phase transitions in a complex network

Radin, Charles; Sadun, Lorenzo

doi:10.1088/1751-8113/46/30/305002

Cited by 68 publications

(158 citation statements)

References 16 publications

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“…For the edge-(multiple)-star model, the desirable star feature relates to network expansiveness and has made predictions of similar asymptotic phenomena possible in broader parameter regions. Related results may be found in Häggström and Jonasson [14], Park and Newman [21], Bianconi [5], Lubetzky and Zhao [20], Radin and Sadun [22,23], and Kenyon et al [16].…”

Section: Introductionsupporting

confidence: 64%

A Detailed Investigation into Near Degenerate Exponential Random Graphs

Yin

2016

J Stat Phys

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The exponential family of random graphs has been a topic of continued research interest. Despite the relative simplicity, these models capture a variety of interesting features displayed by large-scale networks and allow us to better understand how phases transition between one another as tuning parameters vary. As the parameters cross certain lines, the model asymptotically transitions from a very sparse graph to a very dense graph, completely skipping all intermediate structures. We delve deeper into this near degenerate tendency and give an explicit characterization of the asymptotic graph structure as a function of the parameters.

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Section: Introductionsupporting

confidence: 64%

A Detailed Investigation into Near Degenerate Exponential Random Graphs

Yin

2016

J Stat Phys

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“…is a local optimizer for the micro model with τ being the triangle density (see Radin and Sadun [16]). By the same analysis as before, we can see that the optimizing graphon is non-uniform for β 2 < β Proposition 10.…”

Section: Attractive Regimementioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Structure of Constrained Exponential Random Graph Models

Zhu

2017

J Stat Phys

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Abstract. In this paper, we study exponential random graph models subject to certain constraints. We obtain some general results about the asymptotic structure of the model. We show that there exists non-trivial regions in the phase plane where the asymptotic structure is uniform and there also exists non-trivial regions in the phase plane where the asymptotic structure is nonuniform. We will get more refined results for the star model and in particular the two-star model for which a sharp transition from uniform to non-uniform structure, a stationary point and phase transitions will be obtained.

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“…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.…”

Section: Exponential Random Graphsmentioning

confidence: 99%

An introduction to large deviations for random graphs

Chatterjee

2016

Bull. Amer. Math. Soc.

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Abstract. This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the theory of large deviations. This is followed by a description of some large deviation questions about random graphs and an outline of the recent progress on this topic. A more elaborate discussion follows, with a brief account of graph limit theory and its application in constructing a large deviation theory for dense random graphs. The role of Szemerédi's regularity lemma is explained, together with a sketch of the proof of the main large deviation result and some examples. Applications to exponential random graph models are briefly touched upon. The remainder of the paper is devoted to large deviations for sparse graphs. Since the regularity lemma is not applicable in the sparse regime, new tools are needed. Fortunately, there have been several new breakthroughs that managed to achieve the goal by an indirect method. These are discussed, together with an exposition of the underlying theory. The last section contains a list of open problems.

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Phase transitions in a complex network

Cited by 68 publications

References 16 publications

A Detailed Investigation into Near Degenerate Exponential Random Graphs

A Detailed Investigation into Near Degenerate Exponential Random Graphs

Asymptotic Structure of Constrained Exponential Random Graph Models

An introduction to large deviations for random graphs

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