63,373 research outputs found
Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
Spectra of sparse non-Hermitian random matrices determine the dynamics of
complex processes on graphs. Eigenvalue outliers in the spectrum are of
particular interest, since they determine the stationary state and the
stability of dynamical processes. We present a general and exact theory for the
eigenvalue outliers of random matrices with a local tree structure. For
adjacency and Laplacian matrices of oriented random graphs, we derive
analytical expressions for the eigenvalue outliers, the first moments of the
distribution of eigenvector elements associated with an outlier, the support of
the spectral density, and the spectral gap. We show that these spectral
observables obey universal expressions, which hold for a broad class of
oriented random matrices.Comment: 25 pages, 4 figure
The spectral analysis of random graph matrices
A random graph model is a set of graphs together with a probability distribution on that set. A random matrix is a matrix with entries consisting of random values from some specified distribution. Many different random matrices can be associated with a random graph. The spectra of these corresponding matrices are called the spectra of the random graph. The spectra of random graphs are critical to understanding the properties of random graphs. This thesis contains a number of results on the spectra and related spectral properties of several random graph models. In Chapter 1, we briefly present the background, some history as well as the main ideas behind our work. Apart from the introduction in Chapter 1, the first part of the main body of the thesis is Chapter 2. In this part we estimate the eigenvalues of the Laplacian matrix of random multipartite graphs. We also investigate some spectral properties of random multipartite graphs, such as the Laplacian energy, the Laplacian Estrada index, and the von Neumann entropy. The second part consists of Chapters 3, 4, 5 and 6. Guo and Mohar showed that mixed graphs are equivalent to digraphs if we regard (replace) each undirected edge as (by) two oppositely directed arcs. Motivated by the work of Guo and Mohar, we initially propose a new random graph model – the random mixed graph. Each arc is determined by an in-dependent random variable. The main themes of the second part are the spectra and related spectral properties of random mixed graphs. In Chapter 3, we prove that the empirical distribution of the eigenvalues of the Hermitian adjacency matrix converges to Wigner’s semicircle law. As an application of the LSD of Hermitian adjacency matrices, we estimate the Hermitian energy of a random mixed graph. In Chapter 4, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We derive a separation result between the first and the remaining eigenvalues of the Hermitian adjacency matrix. As an application of the asymptotic behavior of the spectrum of the Hermitian adjacency matrix, we estimate the spectral moments of random mixed graphs. In Chapter 5, we prove that the empirical distribution of the eigenvalues of the normalized Hermitian Laplacian matrix converges to Wigner’s semicircle law. Moreover, in Chapter 6, we provide several results on the spectra of general random mixed graphs. In particular, we present a new probability inequality for sums of independent, random, self-adjoint matrices, and then apply this probability inequality to matrices arising from the study of random mixed graphs. We prove a concentration result involving the spectral norm of a random matrix and that of its expectation. Assuming that the probabilities of all the arcs of the mixed graph are mutually independent, we write the Hermitian adjacency matrix as a sum of random self-adjoint matrices. Using this, we estimate the spectrum of the Hermitian adjacency matrix, and prove a concentration result involving the spectrum of the normalized Hermitian Laplacian matrix and its expectation. Finally, in Chapter 7, we estimate upper bounds for the spectral radii of the skew adjacency matrix and skew Randić matrix of random oriented graphs
Random graph states, maximal flow and Fuss-Catalan distributions
For any graph consisting of vertices and edges we construct an
ensemble of random pure quantum states which describe a system composed of
subsystems. Each edge of the graph represents a bi-partite, maximally entangled
state. Each vertex represents a random unitary matrix generated according to
the Haar measure, which describes the coupling between subsystems. Dividing all
subsystems into two parts, one may study entanglement with respect to this
partition. A general technique to derive an expression for the average
entanglement entropy of random pure states associated to a given graph is
presented. Our technique relies on Weingarten calculus and flow problems. We
analyze statistical properties of spectra of such random density matrices and
show for which cases they are described by the free Poissonian
(Marchenko-Pastur) distribution. We derive a discrete family of generalized,
Fuss-Catalan distributions and explicitly construct graphs which lead to
ensembles of random states characterized by these novel distributions of
eigenvalues.Comment: 37 pages, 24 figure
Boosted Simon-Wolff Spectral Criterion and Resonant Delocalization
Discussed here are criteria for the existence of continuous components in the
spectra of operators with random potential. First, the essential condition for
the Simon-Wolff criterion is shown to be measurable at infinity. By
implication, for the iid case and more generally potentials with the K-property
the criterion is boosted by a zero-one law. The boosted criterion, combined
with tunneling estimates, is then applied for sufficiency conditions for the
presence of continuous spectrum for random Schr\"odinger operators. The general
proof strategy which this yields is modeled on the resonant delocalization
arguments by which continuous spectrum in the presence of disorder was
previously established for random operators on tree graphs. In another
application of the Simon-Wolff rank-one analysis we prove the almost sure
simplicity of the pure point spectrum for operators with random potentials of
conditionally continuous distribution.Comment: In version 2 the presentation was somewhat streamlined, and a related
new (/improved) result was added (Appendix B
Exploring the “Middle Earth” of network spectra via a Gaussian matrix function
We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrodinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index - an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patters, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs
Spectra of complex networks
We propose a general approach to the description of spectra of complex
networks. For the spectra of networks with uncorrelated vertices (and a local
tree-like structure), exact equations are derived. These equations are
generalized to the case of networks with correlations between neighboring
vertices. The tail of the density of eigenvalues at large
is related to the behavior of the vertex degree distribution
at large . In particular, as , . We propose a simple approximation, which enables us to
calculate spectra of various graphs analytically. We analyse spectra of various
complex networks and discuss the role of vertices of low degree. We show that
spectra of locally tree-like random graphs may serve as a starting point in the
analysis of spectral properties of real-world networks, e.g., of the Internet.Comment: 10 pages, 4 figure
Spectra of Sparse Random Matrices
We compute the spectral density for ensembles of of sparse symmetric random
matrices using replica, managing to circumvent difficulties that have been
encountered in earlier approaches along the lines first suggested in a seminal
paper by Rodgers and Bray. Due attention is payed to the issue of localization.
Our approach is not restricted to matrices defined on graphs with Poissonian
degree distribution. Matrices defined on regular random graphs or on scale-free
graphs, are easily handled. We also look at matrices with row constraints such
as discrete graph Laplacians. Our approach naturally allows to unfold the total
density of states into contributions coming from vertices of different local
coordination.Comment: 22 papges, 8 figures (one on graph-Laplacians added), one reference
added, some typos eliminate
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