Related papers: Planar Novikov-Shubin invariant for adjacency matr…
Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific…
We give a polynomial bound on the spectral density function of a matrix over the complex group ring of Z^d. It yields an explicit lower bound on the Novikov-Shubin invariant associated to this matrix showing in particular that the…
We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of…
We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit…
Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction…
Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density…
In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the…
We present classes of models in which particles are dropped on an arbitrary fixed finite connected graph, obeying adhesion rules with screening. We prove that there is an invariant distribution for the resulting height profile, and Gaussian…
In this paper, we study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models. The parameter matrix of such a model encodes a directed graph whose vertices correspond to the…
We consider the adjacency matrix associated with a graph that describes transitions between $2^{N}$ states of the discrete Preisach memory model. This matrix can also be associated with the last-in-first-out inventory management rule. We…
We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small…
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…
In this work, we characterise the statistics of Markov chains by constructing an associated sequence of periodic differential operators. Studying the density of states of these operators reveals the absolutely continuous invariant measure…
We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the…
We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.
We review the application of the notion of local convergence on locally finite randomly rooted graphs, known as Benjamini-Schramm convergence, to the calculation of the global eigenvalue density of random matrices from the beta-Gaussian and…
In this paper we construct a two-parameter version of spectral density functions and Novikov-Shubin invariants on fibre bundles. The aim of this approach is to gain a better understanding of how the near-zero spectrum of the Hodge Laplace…
The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants…
We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For…
Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behavior and fitting properties are still critical challenges, that in general, require model specific…