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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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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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
- Random graphs
- Stochastic blockmodels
- Asymptotic normality
- Eigenvalues distribution.
AMS (2000) subject classification
- Primary 62H12
- Secondary 05C50