Related papers: On the extremal spectral properties of random grap…

Desirable random graph models (RGMs) should (i) reproduce common patterns in real-world graphs (e.g., power-law degrees, small diameters, and high clustering), (ii) generate variable (i.e., not overly similar) graphs, and (iii) remain…

We consider the bulk eigenvalue statistics of Laplacian matrices of large Erd\H{o}s-R\'enyi random graphs in the regime $p \geq N^{\delta}/N$ for any fixed $\delta >0$. We prove a local law down to the optimal scale $\eta \gtrsim N^{-1}$…

We analyse the eigenvectors of the adjacency matrix of a random inhomogeneous graph constructed from a specified degree sequence. We assume that the empirical degree sequence has bounded mean and variance. We show that near the edges of the…

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…

The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown…

The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random…

Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric…

We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…

Let $G=G(n,p_n)$ be a homogeneous Erd\"os-R\'enyi graph, and $A$ its adjacency matrix with eigenvalues $\lambda_1(A) \geq \lambda_2(A) \geq ... \geq \lambda_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit…

We consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R\'enyi graph $G(N, p)$. We show that the fluctuations of the…

In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erd\H{o}s-R\'enyi random graph $\mathcal{G}_{n,p}$, in the regime of $p$ where the edge of the spectrum is no…

We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world and random networks follow…

Inhomogeneous Erd\H{o}s-R\'enyi random graphs $\mathbb G_N$ on $N$ vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices $\{i,j\}$ is retained with probability…

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the…

We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The…

The extreme eigenvalues of adjacency matrices are important indicators on the influences of topological structures to collective dynamical behavior of complex networks. Recent findings on the ensemble averageability of the extreme…

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We…

We confirm the long-standing prediction that $c=e\approx 2.718$ is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erd\H{o}s-Renyi random graph with…

Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behavior and fitting properties are still critical challenges, that in general, require model specific…

In this paper, we study the dynamics of a viral spreading process in random geometric graphs (RGG). The spreading of the viral process we consider in this paper is closely related with the eigenvalues of the adjacency matrix of the graph.…