Related papers: Eigenvalue Statistics of One-Face Maps
We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an…
Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…
The Google matrix is a positive, column-stochastic matrix that is used to compute the pagerank of all the web pages on the Internet: the eigenvector corresponding to the eigenvalue 1 is the pagerank vector. Due to its huge dimension, of the…
In this article, we analyze the limiting eigenvalue distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing $n$ nodes on the $d$-dimensional torus $\mathbb{T}^d \equiv [0, 1]^d$ and connecting…
In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family of trees with distance eigenvalue $-1$, and determine all $\{C_3,C_4\}$-free connected graphs with…
We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…
Motivated by the intriguing behavior displayed in a dynamic network that models a population of extreme introverts and extroverts (XIE), we consider the spectral properties of ensembles of random split graph adjacency matrices. We discover…
A graph is said to be I-eigenvalue free if it has no eigenvalues in the interval I with respect to the adjacency matrix A. In this paper we present two algorithms for generating I-eigenvalue free threshold graphs.
The distribution of eigenvalues of N times N random matrices in the limit N to infinity is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a…
We discuss probabilistic models of random covariance structures defined by distributions over sparse eigenmatrices. The decomposition of orthogonal matrices in terms of Givens rotations defines a natural, interpretable framework for…
We consider extremal eigenvalues of sparse random matrices, a class of random matrices including the adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs $\mathcal{G}(N,p)$. Recently, it was shown that the leading order fluctuations of extremal…
Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is…
We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar…
The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically…
We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of \cite{S:08} on the representation of the…
Using exact numerical diagonalization, we investigate localization in two classes of random matrices corresponding to random graphs. The first class comprises the adjacency matrices of Erdos-Renyi (ER) random graphs. The second one…
Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions…
This paper systematically studies the behavior of the leading eigenvectors for independent edge undirected random graphs generated from a general latent position model whose link function is possibly infinite rank and also possibly…
Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a…
In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we…