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

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.

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Funding

The research leading to these results received funding from the Defense Advanced Research Projects Agency under Grant Agreement No. FA8750-12-2-0303.

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Correspondence to Avanti Athreya.

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Athreya, A., Cape, J. & Tang, M. Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00268-x

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Keywords

  • Random graphs
  • Stochastic blockmodels
  • Asymptotic normality
  • Eigenvalues distribution.

AMS (2000) subject classification

  • Primary 62H12
  • Secondary 05C50
  • 62F12.