Related papers: Eigenvectors of the discrete Laplacian on regular …
This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian…
In this work, we study some statistical properties of the extreme eigenstates of the randomly-weighted adjacency matrices of random graphs. We focus on two random graph models: Erd\H{o}s-R\'{e}nyi (ER) graphs and random geometric graphs…
We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. Our results apply to a generalized $N\times N$ Rosenzweig-Porter model. We prove that the eigenvectors…
The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many…
We describe extensive computational experiments on spectral properties of random objects - random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We…
Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected…
We discuss probabilistic models of random covariance structures defined by distributions over sparse eigenmatrices. The decomposition of orthogonal matrices in terms of Givens rotations defines a natural, interpretable framework for…
In this paper, we investigate some relations between the invariants (including vertex and edge connectivity and forwarding indices) of a graph and its Laplacian eigenvalues. In addition, we present a sufficient condition for the existence…
Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…
We show that correlation matrices with particular average and variance of the correlation coefficients have a notably restricted spectral structure. Applying geometric methods, we derive lower bounds for the largest eigenvalue and the…
Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable…
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…
An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number $\beta$.…
Random models of evolution are instrumental in extracting rates of microscopic evolutionary mechanisms from empirical observations on genetic variation in genome sequences. In this context it is necessary to know the statistical properties…
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…
Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
We consider highly heterogeneous random networks with symmetric interactions in the limit of high connectivity. A key feature of this system is that the spectral density of the corresponding ensemble exhibits a divergence within the bulk.…
Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with…
We consider the normalized Laplace operator for directed graphs with positive and negative edge weights. This generalization of the normalized Laplace operator for undirected graphs is used to characterize directed acyclic graphs. Moreover,…