Related papers: Universality in Complex Networks: Random Matrix An…
Graphical models are commonly used to represent conditional dependence relationships between variables. There are multiple methods available for exploring them from high-dimensional data, but almost all of them rely on the assumption that…
Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the…
For the correlated Gaussian Wishart ensemble we compute the distribution of the smallest eigenvalue and a related gap probability.We obtain exact results for the complex (\beta=2) and for the real case (\beta=1). For a particular set of…
The extreme eigenvalues of connectivity matrices govern the influence of the network structure on a number of network dynamical processes. A fundamental open question is whether the eigenvalues of large networks are well represented by…
Theoretical analysis of biological and artificial neural networks e.g. modelling of synaptic or weight matrices necessitate consideration of the generic real-asymmetric matrix ensembles, those with varying order of matrix elements e.g. a…
Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however,…
Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current…
The Krakow-Orsay collaboration has applied methods borrowed from equilibrium statistical mechanics and analytic combinatorics to study the geometry of random networks. Results contained in a series of recent publications and concerning…
In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably…
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…
Extremal spacings between eigenvalues of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N = 4. We study…
We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution…
The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analyzed. In non-integrable regimes we…
Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum, networks are used to describe a wide range of phenomena, such as phase transitions, intricate…
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…
We study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space. We let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this…
In statistical mechanical investigations on complex networks, it is useful to employ random graphs ensembles as null models, to compare with experimental realizations. Motivated by transcription networks, we present here a simple way to…
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble,…
Bayesian networks provide a method of representing conditional independence between random variables and computing the probability distributions associated with these random variables. In this paper, we extend Bayesian network structures to…
In this paper we consider Wigner random matrices -- symmetric n by n random matrices whose entries are independent identically distributed real random variables. We prove that the probability distribution of one or several eigenvalues close…