A Detailed Investigation into Near Degenerate Exponential Random Graphs

Yin, Mei

doi:10.1007/s10955-016-1539-3

Cited by 6 publications

(6 citation statements)

References 32 publications

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“…The complete clique is simply no longer an allowed configuration, and hence the number of triangles becomes fully tuneable, if the model parameters scale appropriately with the system size. In addition, in [25] it is shown that the 'soft' version of our model would have a phase diagram reminiscent of ours. In both cases the sign of a linear combination of functions of the parameters determines the phase of the ensemble.…”

Section: Discussionsupporting

confidence: 53%

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Exactly solvable random graph ensemble with extensively many short cycles

López

Barucca

Fekom

et al. 2018

J. Phys. A: Math. Theor.

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Abstract. We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

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Section: Discussionsupporting

confidence: 53%

“…This abrupt transition was found to be a generic feature of exponential random graph models. As was shown in [24,25], this phenomenon will be observed not only in twoparameter models like the Strauss model, but in any exponential graph ensemble that is biased such as to induce a finite number of subgraph densities.…”

Section: Introductionmentioning

confidence: 70%

Exactly solvable random graph ensemble with extensively many short cycles

López

Barucca

Fekom

et al. 2018

J. Phys. A: Math. Theor.

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“…The theory of graph limits developed by Lovász and coauthors [22] has received phenomenal attention over the last few years. It connects various topics such as graph homomorphisms, Szemeredi's regularity lemma, and quasirandom graphs, and has found many interesting applications in statistical physics, extremal graph theory, statistics and related areas (see [5,6,13,25,26,29,30] and the references therein). Here we recall the basic definitions about the convergence of graph sequences.…”

Section: 21mentioning

confidence: 99%

Replica Symmetry in Upper Tails of Mean-Field Hypergraphs

Mukherjee¹,

Bhattacharya²

2018

Preprint

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Given a sequence of s-uniform hypergraphs {Hn} n≥1 , denote by Tp(Hn) the number of edges in the random induced hypergraph obtained by including every vertex in Hn independently with probability p ∈ (0, 1). Recent advances in the large deviations of low complexity non-linear functions of independent Bernoulli variables can be used to show that tail probabilities of Tp(Hn) are precisely approximated by the so-called 'mean-field' variational problem, under certain assumptions on the sequence {Hn} n≥1 . In this paper, we study properties of this variational problem for the upper tail of Tp(Hn), assuming that the mean-field approximation holds. In particular, we show that the variational problem has a universal replica symmetric phase (where it is uniquely minimized by a constant function), for any sequence of regular s-uniform hypergraphs, which depends only on s.We also analyze the associated variational problem for the related problem of estimating subgraph frequencies in a converging sequence of dense graphs. Here, the variational problems themselves have a limit which can be expressed in terms of the limiting graphon.

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“…exponential ensemble of [10]. Unfortunately, upon varying the control parameters this ensemble was found to switch between very weak clustering and dominance by dense graphs [11][12][13]. See also [14].…”

Section: Introductionmentioning

confidence: 99%

Imaginary replica analysis of loopy regular random graphs

López¹,

Coolen²

2020

J. Phys. A: Math. Theor.

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We present an analytical approach for describing spectrally constrained maximum entropy ensembles of finitely connected regular loopy graphs, valid in the regime of weak loop-loop interactions. We derive an expression for the leading two orders of the expected eigenvalue spectrum, through the use of infinitely many replica indices taking imaginary values. We apply the method to models in which the spectral constraint reduces to a soft constraint on the number of triangles, which exhibit 'shattering' transitions to phases with extensively many disconnected cliques, to models with controlled numbers of triangles and squares, and to models where the spectral constraint reduces to a count of the number of adjacency matrix eigenvalues in a given interval. Our predictions are supported by MCMC simulations based on edge swaps with nontrivial acceptance probabilities.

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A Detailed Investigation into Near Degenerate Exponential Random Graphs

Cited by 6 publications

References 32 publications

Exactly solvable random graph ensemble with extensively many short cycles

Exactly solvable random graph ensemble with extensively many short cycles

Replica Symmetry in Upper Tails of Mean-Field Hypergraphs

Imaginary replica analysis of loopy regular random graphs

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