Related papers: Random matrix analysis of network Laplacians
This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional…
Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper…
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…
Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the…
Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations of positive-definite matrices (here called covariance…
Can a graph specifying the pattern of connections of a dynamical network be reconstructed from statistical properties of a signal generated by such a system? In this model study, we present an evolutionary algorithm for reconstruction of…
Complex networks has been a hot topic of research over the past several years over crossing many disciplines, starting from mathematics and computer science and ending by the social and biological sciences. Random graphs were studied to…
Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…
We study the distribution of the minimum spacing between eigenvalues of a random n by n unitary matrix. The minimum spacing scales as $n^{-4/3}$, not $n^{-2}$ as would be the case for n independent points on the unit circle, illustrating…
We investigate the local spectral statistics of the loss surface Hessians of artificial neural networks, where we discover excellent agreement with Gaussian Orthogonal Ensemble statistics across several network architectures and datasets.…
Models of network diffusion typically rely on the Laplacian matrix, capturing interactions via direct connections. Beyond direct interactions, information in many systems can also flow via indirect pathways, where influence typically…
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…
We develop a statistical theory of networks. A network is a set of vertices and links given by its adjacency matrix $\c$, and the relevant statistical ensembles are defined in terms of a partition function $Z=\sum_{\c} \exp {[}-\beta \H(\c)…
We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert--Schmidt inner product) within a real-linear subspace of the space of $n\times n$ matrices. The matrices we…
A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The…
Here, using two real non-zero parameters $\lambda$ and $\mu$, we construct Gaussian pseudo-orthogonal ensembles of a large number $N$ of $n \times n$ ($n$ even and large) real pseudo-symmetric matrices under the metric $\eta$ using $…
Network topology inference is a fundamental problem in many applications of network science, such as locating the source of fake news, brain connectivity networks detection, etc. Many real-world situations suffer from a critical problem…
In graph signal processing, the graph adjacency matrix or the graph Laplacian commonly define the shift operator. The spectral decomposition of the shift operator plays an important role in that the eigenvalues represent frequencies and the…
Recently, Watts and Strogatz introduced the so-called small-world networks in order to describe systems which combine simultaneously properties of regular and of random lattices. In this work we study diffusion processes defined on such…
Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones…