Related papers: Spectra of Modular and Small-World Matrices
Spectral embedding of adjacency or Laplacian matrices of undirected graphs is a common technique for representing a network in a lower dimensional latent space, with optimal theoretical guarantees. The embedding can be used to estimate the…
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and…
We compute spectra of large stochastic matrices $W$, defined on sparse random graphs, where edges $(i,j)$ of the graph are given positive random weights $W_{ij}>0$ in such a fashion that column sums are normalized to one. We compute spectra…
We study random graphs with possibly different edge probabilities in the challenging sparse regime of bounded expected degrees. Unlike in the dense case, neither the graph adjacency matrix nor its Laplacian concentrate around their…
Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the…
In this article, we study random graphs with a given degree sequence $d_1, d_2, \cdots, d_n$ from the configuration model. We show that under mild assumptions of the degree sequence, the spectral distribution of the normalized Laplacian…
We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the…
Using spectral embedding based on the signless Laplacian, we obtain bounds on the spectrum of transition matrices on graphs. As a consequence, we bound return probabilities and the uniform mixing time of simple random walk on graphs. In…
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…
Collective dynamics on small-world networks emerge in a broad range of systems with their spectra characterizing fundamental asymptotic features. Here we derive analytic mean field predictions for the spectra of small-world models that…
In this work, we develop a spectral theory for hypergraph limits. We prove the convergence of the spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the $1$-cut metric. On the other hand, we give examples of…
We quantify the dynamical implications of the small-world phenomenon. We consider the generic synchronization of oscillator networks of arbitrary topology, and link the linear stability of the synchronous state to an algebraic condition of…
We propose a novel model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the…
In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state…
Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph…
A method of resummation of infinite series of perturbation theory diagrams is applied for studying the properties of random band matrices. The topological classification of Feynman diagrams, which was actively used in last years for matrix…
Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of…
We study structure, eigenvalue spectra and diffusion dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their…
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…
We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and…