Native ultrametricity of sparse random ensembles

Аветисов, В. А.; Krapivsky, P. L.; Nechaev, Sergei

doi:10.1088/1751-8113/49/3/035101

Cited by 12 publications

(25 citation statements)

References 64 publications

(122 reference statements)

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“…(c is some positive constant) and is, as shown in [11] for sparse ensembles, a manifestation of a Lifshitz tail for the Andersen localization.…”

mentioning

confidence: 91%

“…In the quenched model, within the transition region, isolated eigenvalues of A form the second zone in the spectrum and correspond one-by-one to clusters in the large network (see [8][9][10] for general description). Above the transition point, the spectral density (SD), ρ(λ), of the adjacency matrix of each clique (almost fully connected subgraph) is the same as the spectral density of the sparse matrix, and has Lifshitz tails typical for 1D Anderson localization, as discussed in [11]. The spectral density of the whole network has a triangle-like shape typically seen in scale-free networks.…”

mentioning

confidence: 94%

“…The triangle-shaped SD is typical for the scalefree networks [12,13], however in our case the vertex degree is fixed at network preparation and can be only redistributed between cliques. The remarkable point is that the SD evaluated for each particular clique exhibits a hierarchical set of resonance peaks typical for sparse matrices, which have been analytically investigated in [11].…”

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confidence: 99%

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Eigenvalue tunneling and decay of quenched random network

et al. 2016

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We consider the canonical ensemble of N -vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity µ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p −1 ] almost full subgraphs (cliques) above critical fugacity, µc, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing µ leads to the formation of two-zonal support for µ > µc. Eigenvalue tunnelling from one (central) zone to the other means formation of a new clique in the defragmentation process. The adjacency matrix of the ground state of a network has the block-diagonal form where number of vertices in blocks fluctuate around the mean value N p. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.Investigation of critical and collective effects in graphs and networks has becoming a new rapidly developing interdisciplinary area, with diverse applications and variety of questions to be asked, see [1] for review. Ensembles of random Erdős-Rényi topological graphs (networks) provide an efficient laboratory for testing collective phenomena in statistical physics of complex systems, being also tightly linked to conventional random matrix theory. Besides investigating typical statistical properties of networks, like vertex degree distribution, clustering coefficients, "small world" structure etc, last two decades have been marked by rapidly growing interest in more refined graph characteristics, such as distribution of small subgraphs involving triads of vertices.Triadic interactions, being the simplest interactions beyond the free-field theory, play crucial role in the network statistics. Presence of such interactions is responsible for emergence of phase transitions in complex distributed systems. First example of a phase transition in random networks, known as Strauss clustering [2], has been treated by the Random Matrix Theory (RMT) in [3]. It was argued that, when the increasing fugacity, µ, the system develops two phases with essentially different triad concentrations. At large µ the system falls into the Strauss phase with the single clique of nodes. The condensation of triads is a non-perturbative phenomenon identified in [4] with the 1st order phase transition in the framework of mean-field cavity-like approach.Similar critical behavior was found in [5] for the vertexdegree-conserved ER graphs. It was demonstrated in the framework of the mean-field approach that the phase transition takes place in this case as well. The hysteresis for dependence of the triad concentration on the fugacity, µ also has been observed in [5]. For bi-color networks with conserved vertex degree a new phenomena of a wide plateau formation in concentration of black-white bonds as a function of the fugacity of unicolor triples of bonds has been found in...

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“…(c is some positive constant) and is, as shown in [11] for sparse ensembles, a manifestation of a Lifshitz tail for the Andersen localization.…”

mentioning

confidence: 91%

mentioning

confidence: 94%

mentioning

confidence: 99%

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Eigenvalue tunneling and decay of quenched random network

et al. 2016

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“…As concerns exact results, we have shown in Ref. [48] that the spectral density, ρ lin (λ), in the ensemble of linear chains with the exponential distribution of their lengths demonstrates a very peculiar ultrametric structure related to the so-called "popcorn function" [49].…”

Section: Rare-event Statistics and Hyperbolic Geometrymentioning

confidence: 83%

“…In Ref. [48] an explicit expression for Q n has been derived within a kinetic theory approach which allows one to determine the size distribution of generic clusters (irrespective of their topology) and then adopt this framework to the computation of the size distribution of linear chains. As concerns exact results, we have shown in Ref.…”

Section: Rare-event Statistics and Hyperbolic Geometrymentioning

confidence: 99%

Non-Euclidean Geometry in Nature

Nechaev

2017

Order, Disorder and Criticality

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I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random graphs. For these systems I discuss the following interlinked questions: (i) the optimal embedding of plants leaves in the three-dimensional space, (ii) the spectral statistics of sparse random matrix ensembles.

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Lifshitz tails at spectral edge and holography with a finite cutoff

Gorsky

Nechaev

Valov

2021

J. High Energ. Phys.

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We propose the holographic description of the Lifshitz tail typical for one-particle spectral density of bounded disordered system in D = 1 space. To this aim the “polymer representation” of the Jackiw-Teitelboim (JT) 2D dilaton gravity at a finite cutoff is used and the corresponding partition function is considered as the weighted sum over paths of fixed length in an external magnetic field. We identify the regime of small loops, responsible for emergence of a Lifshitz tail in the Gaussian disorder, and relate the strength of disorder to the boundary value of the dilaton. The geometry corresponding to the Poisson disorder in the boundary theory involves random paths fluctuating in the vicinity of the hard impenetrable cut-off disc in a 2D plane. It is shown that the ensemble of “stretched” paths evading the disc possesses the Kardar-Parisi-Zhang (KPZ) scaling for fluctuations, which is the key property that ensures the dual description of the Lifshitz tail in the spectral density for the Poisson disorder.

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Native ultrametricity of sparse random ensembles

Cited by 12 publications

References 64 publications

Eigenvalue tunneling and decay of quenched random network

Eigenvalue tunneling and decay of quenched random network

Non-Euclidean Geometry in Nature

Lifshitz tails at spectral edge and holography with a finite cutoff

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