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Characterization of Random Matrix Eigenvectors for Stochastic Block Model

Abstract : The eigenvalue spectrum of the adjacency matrix of Stochastic Block Model (SBM) consists of two parts: a finite discrete set of dominant eigenvalues and a continuous bulk of eigenvalues. We characterize analytically the eigenvectors corresponding to the continuous part: the bulk eigenvectors. For symmetric SBM adjacency matrices, the eigenvectors are shown to satisfy two key properties. A modified spectral function of the eigenvalues, depending on the eigenvectors, converges to the eigenvalue spectrum. Its fluctuations around this limit converge to a Gaussian process different from a Brownian bridge. This latter fact disproves that the bulk eigenvectors are Haar distributed.
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Contributor : Konstantin Avrachenkov <>
Submitted on : Tuesday, January 19, 2016 - 12:46:24 PM
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Arun Kadavankandy, Laura Cottatellucci, Konstantin Avrachenkov. Characterization of Random Matrix Eigenvectors for Stochastic Block Model. 49th Asilomar Conference on Signals, Systems, and Computer, Nov 2015, Pacific Grove, CA, United States. ⟨hal-01258664⟩



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