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A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…
We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small…
We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random…
The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the…
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…
Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudo-unitary group. Further, we develop a random matrix theory which is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices as…
Sparse graphs built by sparse representation has been demonstrated to be effective in clustering high-dimensional data. Albeit the compelling empirical performance, the vanilla sparse graph ignores the geometric information of the data by…
We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a…
Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we…
Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes.…
We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk…
For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…
Let $G=(V,E)$ be a $d$-regular graph on $n$ vertices and let $\mu_0$ be a probability measure on $V$. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on $V$ given by $\mu_{k+1} = A…
Random $s$-intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random $s$-intersection graph is constructed by first assigning each vertex a set of items in some random manner,…
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability…
Statistical field theory methods have been very successful with a number of random graph and random matrix problems, but it is challenging to apply these methods to graphs with prescribed degree sequences due to the extensive number of…
Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two…
This work examines various statistical distributions in connection with random Vandermonde matrices and their extension to $d$--dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to…
We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate…