Random incidence matrices: moments of the spectral density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of "small" eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit), we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e=2.72... is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix. Keywords: random graphs, random matrices, sparse matrices, incidence matrices spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified

Real symmetric random matrices and paths counting

Exact evaluation of $$ is here performed for real symmetric matrices $S$ of arbitrary order $n$, up to some integer $p$, where the matrix entries are independent identically distributed random variables, with an arbitrary probability distribution. These expectations are polynomials in the moments of the matrix entries ; they provide useful information on the spectral density of the ensemble in the large $n$ limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large $n$ limit.Comment: 23 pages, 10 figures, revised pape

On Connected Diagrams and Cumulants of Erdos-Renyi Matrix Models

Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdos-Renyi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogs of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions in the cases of constant and vanishing edge probabilities as n tends to infinity.Comment: 34 pages, 8 figure

Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edge between the nodes $\{i,j\}$ and zero otherwise, and consider the normalized random adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $[\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution of $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empirical measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the symmetric matrix $\mathbf{A}_n$. Under some technical conditions on the expected degree sequence, we show that with probability one, $\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministic distribution $F(\mathord{\cdot})$. Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of $F(\mathord{\cdot})$. We apply our results to well-known degree distributions, such as power-law and exponential. The asymptotic expressions of the spectral moments in each case provide significant insights about the bulk behavior of the eigenvalue spectrum

Sparse random matrices: the eigenvalue spectrum revisited

We revisit the derivation of the density of states of sparse random matrices. We derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit. Using the iterative scheme introduced by Biroli and Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration. The combination of the two methods yields a very simple simple geometric interpretation of the tails of the spectrum. We test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum.Comment: 18 pages, 7 figures. Accepted version, minor corrections, references adde