Cavity approach to the spectral density of sparse symmetric random matrices

Rogers, Tim; Castillo, Isaac Pérez; Kühn, Reimer; Takeda, Kenta

doi:10.1103/physreve.78.031116

Cited by 132 publications

(284 citation statements)

References 22 publications

Supporting

Mentioning

277

Contrasting

Order By: Relevance

“…However, we expect that the properties, which we will investigate after this, do not depend on the details of the generation schemes. When the support of p(k) is not bounded from the above and values of the entries are kept finite, the first eigenvalue generally diverges as N → ∞ [11,12,13,14,15,16]. To avoid this possibility, we assume that p(k) = 0 for k, which is larger than a certain value, k max , unless infinitesimal entries are assumed.…”

Section: Model Definitionmentioning

confidence: 99%

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

Kabashima¹,

Takahashi²,

Watanabe³

2010

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

Abstract.A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.

show abstract

Section: Model Definitionmentioning

confidence: 99%

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

Kabashima¹,

Takahashi²,

Watanabe³

2010

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

show abstract

“…From a mathematical-physicist point of view, in the frame of RMT, in 1988 Rodgers and Bray [12] proposed an ensemble of sparse random matrices characterized by the connectivity ξ. Since then, several papers have been devoted to analytical and numerical studies of sparse symmetric random matrices (see for example [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Among the most relevant results of these studies we can mention that: (i) in the very sparse limit, ξ → 1, the density of states was found to deviate from the Wigner semicircle law with the appearance of singularities, around and at the band center, and tails beyond the semicircle [12][13][14][15][16][17][18][19][20][21]; (ii) a delocalization transition was found at ξ ≈ 1.4 [14][15][16]22]; (iii) the nearest-neighbor energy level spacing distribution P (s) was found to evolve from the Poisson to the Gaussian Orthogonal Ensemble (GOE) predictions for increasing ξ [11,14,16] (the same transition was reported for the number variance in Ref.…”

Section: Introductionmentioning

confidence: 99%

Universality in the spectral and eigenfunction properties of random networks

et al. 2015

View full text Add to dashboard Cite

By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution P (s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ ≡ αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well P (s) in the transition from α = 0, when the vertices in the network are isolated, to α = 1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

show abstract

“…Methods developed in [25] can be used to efficiently deal with the sparsity of the ensemble of matrices considered in the present letter. Alternatively, one can analyse single large instances using a cavity approach proposed in [26] to obtain the single instance spectral density in terms of variances of single-site marginals.…”

mentioning

confidence: 99%

Spectra of random stochastic matrices and relaxation in complex systems

Kühn

2015

EPL

Self Cite

View full text Add to dashboard Cite

We compute spectra of large stochastic matrices W , defined on sparse random graphs, where edges (i, j) of the graph are given positive random weights Wij > 0 in such a fashion that column sums are normalized to one. We compute spectra of such matrices both in the thermodynamic limit, and for single large instances. The structure of the graphs and the distribution of the non-zero edge weights Wij are largely arbitrary, as long as the mean vertex degree remains finite in the thermodynamic limit and the Wij satisfy a detailed balance condition. Knowing the spectra of stochastic matrices is tantamount to knowing the complete spectrum of relaxation times of stochastic processes described by them, so our results should have many interesting applications for the description of relaxation in complex systems. Our approach allows to disentangle contributions to the spectral density related to extended and localized states, respectively, allowing to differentiate between time-scales associated with transport processes and those associated with the dynamics of local rearrangements. There are numerous processes, both natural and artificial, which can be understood in terms of random walks on complex networks [1][2][3], including the spread of diseases in social networks [4,5], the transmission of information in communication networks (e.g.[6]), search algorithms [7,8], the out-of-equilibrium dynamics of glassy systems at low temperatures as described in terms of hopping between long-lived states in state space [9][10][11], the dynamics of major conformational changes in macro-molecules [12], or cell-signalling through protein-protein interaction networks [13], to name but a few. For reviews that cover several of these topics, see e.g. [14][15][16].The purpose of the present letter is to use random matrix theory to contribute to the understanding of systems of this type. We compute spectra of transition matrices for discrete Markov chains describing stochastic dynamics in complex systems. We construct these in terms of sparse random graphs in such a way that an edge (i, j) in a graph corresponds to a possible transition j → i, with the edge weight W ij > 0 quantifying the associated transitions probability, requiring i W ij = 1 for all j. We are interested in the limit, where the number N of possible states becomes large, with the average number of possible transitions at each state remaining finite in the thermodynamic limit (N → ∞).Given a time-dependent probability vector p(t) = (p i (t)), we have an evolution equation of the formThe condition W ij ≥ 0 for all (i, j) and the column sum constraint together entail that the spectrum of W is contained in the unit disc of the complex plane, σ(W ) ⊆ {z; |z| ≤ 1}. If W satisfies a detailed balance condition with an equilibrium distribution, p i = p eq i , such that W ij p j = W ji p i for all pairs (i, j), then W can be symmetrized by a similarity transformation -Our main interest here is the relation between eigenvalues of W and relaxation times of the Markov chain it describes...

show abstract

Cavity approach to the spectral density of sparse symmetric random matrices

Cited by 132 publications

References 22 publications

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

Universality in the spectral and eigenfunction properties of random networks

Spectra of random stochastic matrices and relaxation in complex systems

Contact Info

Product

Resources

About