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

Author

Listed:
  • Avanti Athreya

    (Johns Hopkins University)

  • Joshua Cape

    (University of Pittsburgh)

  • Minh Tang

    (North Carolina State University)

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.

Suggested Citation

  • Avanti Athreya & Joshua Cape & Minh Tang, 2022. "Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 36-63, June.
    • Handle: RePEc:spr:sankha:v:84:y:2022:i:1:d:10.1007_s13171-021-00268-x
    DOI: 10.1007/s13171-021-00268-x
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    References listed on IDEAS

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    1. Srijan Sengupta & Yuguo Chen, 2018. "A block model for node popularity in networks with community structure," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(2), pages 365-386, March.
    2. Tomoki Tokuda, 2018. "Statistical test for detecting community structure in real-valued edge-weighted graphs," PLOS ONE, Public Library of Science, vol. 13(3), pages 1-18, March.
    3. Y. Yu & T. Wang & R. J. Samworth, 2015. "A useful variant of the Davis–Kahan theorem for statisticians," Biometrika, Biometrika Trust, vol. 102(2), pages 315-323.
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