Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs

Chakrabarty, Arijit; Hazra, Rajat Subhra; Hollander, Frank den; Sfragara, Matteo

doi:10.1142/s201032632150009x

Cited by 12 publications

(18 citation statements)

References 34 publications

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“…Spectral properties of random graphs have been studied intensively in past years. A non-exhaustive list of key contributions is [11,12,3,4,7,9,10,13,16,25]. Both the adjacency matrix and the Laplacian matrix have been popular.…”

Section: Introductionmentioning

confidence: 99%

A spectral signature of breaking of ensemble equivalence for constrained random graphs

Dionigi

Garlaschelli

Hollander

et al. 2021

Electron. Commun. Probab.

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For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation ('hard constraint'), while the canonical ensemble requires the constraint to be met only on average ('soft constraint'). It is known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the adjacency matrix of the graph. We consider two examples of constraints in the dense regime: (1) fix the degrees of the vertices (= the degree sequence); (2) fix the sum of the degrees of the vertices (= twice the number of edges). Example (1) imposes an extensive number of local constraints and is known to lead to breaking of ensemble equivalence. Example (2) imposes a single global constraint and is known to lead to ensemble equivalence. Our working hypothesis is that breaking of ensemble equivalence corresponds to a non-vanishing difference of the expected values of the largest eigenvalue under the two ensembles. We verify that, in the limit as the size of the graph tends to infinity, the difference between the expected values of the largest eigenvalue in the two ensembles does not vanish for (1) and vanishes for (2). A key tool in our analysis is a transfer method that uses relative entropy to determine whether probabilistic estimates can be carried over from the canonical ensemble to the microcanonical ensemble, and illustrates how breaking of ensemble equivalence may prevent this from being possible.

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Section: Introductionmentioning

confidence: 99%

A spectral signature of breaking of ensemble equivalence for constrained random graphs

Dionigi

Garlaschelli

Hollander

et al. 2021

Electron. Commun. Probab.

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“…Random graphs generated from an inhomogeneous Erdős‐Rényi model

scriptG false(n, false(p_{i j} false) false)

, where edges exist independently with given probabilities p ij is a generalization of the classical Erdős‐Rényi model

scriptG false(n, p false)

. Recently, there are some results on the largest eigenvalue and the spectrum of the Laplacian matrices of inhomogeneous Erdős‐Rényi model random graphs. Many popular graph models arise as special cases of

scriptG false(n, false(p_{i j} false) false)

such as random graphs with given expected degrees , stochastic block models (SBMs) , and W ‐random graphs .…”

Section: Introductionmentioning

confidence: 99%

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.

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“…It is a pertinent question that whether the conclusion involving Marchenko-Pastur law can be replaced by the standard Wigner's semicircle law, w. The measures of the form w ⊠ ρ for some ρ ∈ M + has appeared as the limiting spectral distributions of random matrices (see Anderson and Zeitouni [2008], Chakrabarty et al [2016Chakrabarty et al [ , 2018b), free type W distributions (see Pérez-Abreu and Sakuma [2012]) and in several other places.…”

Section: Some Corollariesmentioning

confidence: 99%

Regular variation and free regular infinitely divisible laws

Chakrabarty

Chakraborty

Hazra

2020

Statistics & Probability Letters

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In this article the relation between the tail behaviours of a free regular infinitely divisible probability measure and its Lévy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. [1979] and our result using the Bercovici-Pata bijection.2010 Mathematics Subject Classification. 60G70; 46L53; 46L54.

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Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs

Cited by 12 publications

References 34 publications

A spectral signature of breaking of ensemble equivalence for constrained random graphs

A spectral signature of breaking of ensemble equivalence for constrained random graphs

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Regular variation and free regular infinitely divisible laws

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