Top Eigenpair Statistics for Weighted Sparse Graphs

Abstract

We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature, we employ the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by population dynamics. This derivation allows us to identify and unpack the individual contributions to the top eigenvector's components coming from nodes of degree $k$. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, and are further cross-checked on the cases of random regular graphs and sparse Markov transition matrices for unbiased random walks.

arXiv comment47 pages, 7 figures. Revised version, accepted for publication in J. Phys. A

References in corpus (35)

• Evolution of networks
• The large deviation approach to statistical mechanics
• The Bethe lattice spin glass revisited
• The onset of synchronization in large networks of coupled oscillators
• Introduction to Random Matrices - Theory and Practice
• Eigenvector statistics in non-Hermitian random matrix ensembles
• Sparse random matrices: the eigenvalue spectrum revisited
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective
• A single defect approximation for localized states on random lattices
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• On the localization transition in symmetric random matrices
• May's Instability in Large Economies
• Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models
• Weighted percolation on directed networks
• Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach
• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Sparse regular random graphs: Spectral density and eigenvectors
• Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach
• Spectra of sparse non-Hermitian random matrices: an analytical solution
• Eigenvector dynamics: general theory and some applications
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Spectra of Modular and Small-World Matrices
• Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation
• Finitely coordinated models for low-temperature phases of amorphous systems
• On the overlaps between eigenvectors of correlated random matrices
• Equivalence of replica and cavity methods for computing spectra of sparse random matrices
• Spectra of Random Stochastic Matrices and Relaxation in Complex Systems
• Estimating the principal components of correlation matrices from all their empirical eigenvectors
• Spectral density of the non-backtracking operator
• Eigenvector statistics of the product of Ginibre matrices
• Fat-tailed distribution derived from first eigenvector of symmetric random sparse matrix

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• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Linear stability analysis for large dynamical systems on directed random graphs
• Localization and universality of eigenvectors in directed random graphs
• Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations
• Instabilities of complex fluids with partially structured and partially random interactions
• Second largest Eigenpair Statistics for Sparse Graphs
• Random sparse generators of Markovian evolution and their spectral properties
• Top eigenpair statistics of diluted Wishart matrices
• Statistics of the non-zero eigenvalues and singular values of low-rank random matrices with non-negative entries