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Sankhya A

, Volume 78, Issue 1, pp 1–18 | Cite as

A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs

  • A. Athreya
  • C. E. Priebe
  • M. Tang
  • V. Lyzinski
  • D. J. Marchette
  • D. L. Sussman
Article

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.

Keywords and phrases.

Random dot product graph Central limit theorem Model-based clustering 

AMS (2000) subject classification.

Primary 62E20 Secondary 05C80 60F05 

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Copyright information

© Indian Statistical Institute 2015

Authors and Affiliations

  • A. Athreya
    • 1
  • C. E. Priebe
    • 1
  • M. Tang
    • 1
  • V. Lyzinski
    • 2
  • D. J. Marchette
    • 3
  • D. L. Sussman
    • 4
  1. 1.Department of Applied Mathematics and StatisticsJohns Hopkins UniversityBaltimoreUSA
  2. 2.Human Language Technology Center of ExcellenceJohns Hopkins UniversityBaltimoreUSA
  3. 3.Naval Surface Warfare CenterDahlgrenUSA
  4. 4.Department of StatisticsHarvard UniversityCambridgeUSA

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