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A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs

Author

Listed:
  • A. Athreya

    () (Johns Hopkins University)

  • C. E. Priebe

    () (Johns Hopkins University)

  • M. Tang

    () (Johns Hopkins University)

  • V. Lyzinski

    () (Johns Hopkins University)

  • D. J. Marchette

    () (Naval Surface Warfare Center)

  • D. L. Sussman

    () (Harvard University)

Abstract

Abstract We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. We use the spectral embedding of the adjacency matrix to construct consistent estimates for the latent positions, and we show that the appropriately scaled differences between the estimated and true latent positions converge to a mixture of Gaussian random variables. We state several corollaries, including an alternate proof of a central limit theorem for the first eigenvector of the adjacency matrix of an Erdos-Rényi random graph.

Suggested Citation

  • A. Athreya & C. E. Priebe & M. Tang & V. Lyzinski & D. J. Marchette & D. L. Sussman, 2016. "A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(1), pages 1-18, February.
  • Handle: RePEc:spr:sankha:v:78:y:2016:i:1:d:10.1007_s13171-015-0071-x
    DOI: 10.1007/s13171-015-0071-x
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    References listed on IDEAS

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    1. D. S. Choi & P. J. Wolfe & E. M. Airoldi, 2012. "Stochastic blockmodels with a growing number of classes," Biometrika, Biometrika Trust, vol. 99(2), pages 273-284.
    2. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    3. Ting Yan & Jinfeng Xu, 2013. "A central limit theorem in the β-model for undirected random graphs with a diverging number of vertices," Biometrika, Biometrika Trust, vol. 100(2), pages 519-524.
    4. Chris Fraley & Adrian E. Raftery, 1999. "MCLUST: Software for Model-Based Cluster Analysis," Journal of Classification, Springer;The Classification Society, vol. 16(2), pages 297-306, July.
    5. 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.
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    1. repec:bla:jorssb:v:79:y:2017:i:5:p:1295-1366 is not listed on IDEAS

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