Localization and universality of eigenvectors in directed random graphs

Abstract

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree fluctuations. For delocalized eigenvectors, we recover the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.

arXiv comment7 pages, 4 figures, 25 pages Supplemental Material

References in corpus (51)

• Statistical mechanics of complex networks
• Random graphs with arbitrary degree distributions and their applications
• Localization transitions in non-Hermitian quantum mechanics
• Statistics of energy levels and eigenfunctions in disordered systems
• Spectral redemption: clustering sparse networks
• Vortex Pinning and Non-Hermitian Quantum Mechanics
• The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
• Localization and Spreading of Diseases in Complex Networks
• Localization and centrality in networks
• Giant strongly connected component of directed networks
• Spectra of complex networks
• Characterizing the dynamical importance of network nodes and links
• Anderson localization on random regular graphs
• Transition to chaos in random neuronal networks
• Non-Hermitean Localization and De-Localization
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Transition to chaos in random networks with cell-type-specific connectivity
• Spectra of Sparse Random Matrices
• Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
• Distinct types of eigenvector localization in networks
• Random incidence matrices: moments of the spectral density
• Spectra of random graphs with arbitrary expected degrees
• Non-Hermitian Localization in Biological Networks
• The effect of population abundances on the stability of large random ecosystems
• Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• On the localization transition in symmetric random matrices
• Eigenvector localization in real networks and its implications for epidemic spreading
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Analytic computation of the Instantaneous Normal Modes spectrum in low density liquids
• Extended States in a Lifshitz Tail Regime for Random Schrödinger Operators on Trees
• Epidemic threshold in directed networks
• Random Matrix Approach to Glassy Physics --- Low Temperatures and Beyond
• Finitely connected vector spin systems with random matrix interactions
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Spacing ratio characterization of the spectra of directed random networks
• Delocalization transition for the Google matrix
• Localization of eigenvectors in random graphs
• On random symmetric matrices with a constraint: the spectral density of random impedance networks
• Linear stability analysis for large dynamical systems on directed random graphs
• First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation
• Finitely coordinated models for low-temperature phases of amorphous systems
• Eigenvalue Repulsion and Eigenfunction Localization in Sparse Non-Hermitian Random Matrices
• Algorithmic detectability threshold of the stochastic block model
• Spectra of Random Stochastic Matrices and Relaxation in Complex Systems
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Statistics of delta peaks in the spectral density of large random trees
• Synchronization in random balanced networks
• Detection thresholds in very sparse matrix completion
• Stability of gene regulatory networks

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