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

Abstract

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.

Anderson Model on Bethe Lattices: Density of States, Localization Properties and Isolated Eigenvalue

“…We shall follow an approach first proposed in ref. 30) for off-diagonal disorder, and begin by rederiving it with a slightly different presentation, related to the discussion of ref. 18).…”

“…2) we find: 4) which are indeed the same as derived in ref. 30). The equations linking the 'messages' G on the various edges of the graph are decoupled from the y's, and corresponds to the recursion equations for the resolvents stated in (2 .…”

Universality in the spectral and eigenfunction properties of random networks

“…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.…”

The First Eigenvalue of (<i>c</i>,<i>d</i>)-Regular Graph

“…Spectral analysis of graphs plays key roles in various fields of mathematical sciences, such as information science, combinatorics, statistics, physics, economics and sociology [2], [3], [5], [8], [10]. This is in general to analyze eigenvalues and eigenvectors of matrices related to graphs expressing certain relations; in particular, the first (largest) eigenvalue and its corresponding eigenvector are important for understanding the typical structure of graphs.…”

“…[5] and references herein); however, the analysis is still not sufficient, in particular, for sparse random matrices that are important for constructing approximate solutions of various combinatorial problems [10]. One of the important questions is to understand the influence of the fluctuation of degrees.…”