Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices

Avanti Athreya (), Joshua Cape () and Minh Tang ()
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Avanti Athreya: Johns Hopkins University
Joshua Cape: University of Pittsburgh
Minh Tang: North Carolina State University

Sankhya A: The Indian Journal of Statistics, 2022, vol. 84, issue 1, No 2, 36-63

Abstract: Abstract We derive the limiting distribution for the outlier eigenvalues of the adjacency matrix for random graphs with independent edges whose edge probability matrices have low-rank structure. We show that when the number of vertices tends to infinity, the leading eigenvalues in magnitude are jointly multivariate Gaussian with bounded covariances. As a special case, this implies a limiting normal distribution for the outlier eigenvalues of stochastic blockmodel graphs and their degree-corrected or mixed-membership variants. Our result extends the classical result of Füredi and Komlós on the fluctuation of the largest eigenvalue for Erdős–Rényi graphs.

Keywords: Random graphs; Stochastic blockmodels; Asymptotic normality; Eigenvalues distribution.; Primary 62H12; Secondary 05C50; 62F12. (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13171-021-00268-x

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