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A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs

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  • Daniel L. Sussman
  • Minh Tang
  • Donniell E. Fishkind
  • Carey E. Priebe

Abstract

We present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel. We use an embedding procedure motivated by the random dot product graph model, a particular example of the latent position model. The embedding associates each node with a vector; these vectors are clustered via minimization of a square error criterion. We prove that this method is consistent for assigning nodes to blocks, as only a negligible number of nodes will be misassigned. We prove consistency of the method for directed and undirected graphs. The consistent block assignment makes possible consistent parameter estimation for a stochastic blockmodel. We extend the result in the setting where the number of blocks grows slowly with the number of nodes. Our method is also computationally feasible even for very large graphs. We compare our method with Laplacian spectral clustering through analysis of simulated data and a graph derived from Wikipedia documents.

Suggested Citation

  • Daniel L. Sussman & Minh Tang & Donniell E. Fishkind & Carey E. Priebe, 2012. "A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1119-1128, September.
    • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:499:p:1119-1128
    DOI: 10.1080/01621459.2012.699795
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    Cited by:

    1. Patrick Rubin‐Delanchy & Joshua Cape & Minh Tang & Carey E. Priebe, 2022. "A statistical interpretation of spectral embedding: The generalised random dot product graph," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1446-1473, September.
    2. Jochmans, Koen, 2024. "Nonparametric Identification And Estimation of Stochastic Block Models From Many Small Networks”," TSE Working Papers 24-1514, Toulouse School of Economics (TSE).
    3. Chung, Jaewon & Bridgeford, Eric & Arroyo, Jesus & Pedigo, Benjamin D. & Saad-Eldin, Ali & Gopalakrishnan, Vivek & Xiang, Liang & Priebe, Carey E. & Vogelstein, Joshua T., 2020. "Statistical Connectomics," OSF Preprints ek4n3, Center for Open Science.
    4. Yoder, Jordan & Chen, Li & Pao, Henry & Bridgeford, Eric & Levin, Keith & Fishkind, Donniell E. & Priebe, Carey & Lyzinski, Vince, 2020. "Vertex nomination: The canonical sampling and the extended spectral nomination schemes," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    5. Stefanos Bennett & Mihai Cucuringu & Gesine Reinert, 2022. "Lead-lag detection and network clustering for multivariate time series with an application to the US equity market," Papers 2201.08283, arXiv.org.

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