Related papers: Modularity spectra, eigen-subspaces, and structure…
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these…
We compute the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a different covariance matrix. Random matrix theory enjoys many applications including sums and products of random…
The expectation and the mean of partitions generated by a cluster ensemble are not unique in general. This issue poses challenges in statistical inference and cluster stability. In this contribution, we state sufficient conditions for…
How can researchers test for heterogeneity in the local structure of a network? In this paper, we present a framework that utilizes random sampling to give subgraphs which are then used in a goodness of fit test to test for heterogeneity.…
We study the expectation of linear eigenvalue statistics of matrix models with any $\beta>0$, assuming that the potential $V$ is a real analytic function and that the corresponding equilibrium measure has a one-interval support. We obtain…
Measures of complex network analysis, such as vertex centrality, have the potential to unveil existing network patterns and behaviors. They contribute to the understanding of networks and their components by analyzing their structural…
Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in $n$ variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic…
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…
We state theoretical properties for $k$-means clustering of Symmetric Positive Definite (SPD) matrices, in a non-Euclidean space, that provides a natural and favourable representation of these data. We then provide a novel application for…
Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These…
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…
Networks are often studied using the eigenvalues of their adjacency matrix, a powerful mathematical tool with a wide range of applications. Since in real systems the exact graph structure is not known, researchers resort to random graphs to…
Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been…
As an indicator of the stability of spectral clustering of an undirected weighted graph into $k$ clusters, the $k$th spectral gap of the graph Laplacian is often considered. The $k$th spectral gap is characterized in this paper as an…
We consider the problem of testing graph cluster structure: given access to a graph $G=(V, E)$, can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph?…
Motivated by the recent demonstration of its use as a tool for the detection and characterization of phase-shape correlations in multivariate time series, we show that eigenvalue decomposition can also be applied to a matrix of indices of…
The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this…
This paper aims to test the number of spikes in a generalized spiked covariance matrix, the spiked eigenvalues of which may be extremely larger or smaller than the non-spiked ones. For a high-dimensional problem, we first propose a general…
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 consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the…