A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs
- 176 Downloads
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
Unable to display preview. Download preview PDF.
- Bickel, P.J. and Chen, A. (2009). A nonparametric view of network models and Newman-Girvan and other modularities. Proc. Natl. Acad. Sci. USA 106, 21, 068–73.Google Scholar
- Chung, F.R.K. (1997). Spectral Graph Theory. American Mathematical Society.Google Scholar
- Hoover, D.N. (1979). Relations on probability spaces and arrays of random variables. Tech. rep. Institute for Advanced Study.Google Scholar
- Knowles, A. and Yin, J. (2011). Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields, 1–40.Google Scholar
- Marchette, D., Priebe, C.E. and Coppersmith, G. (2011). Vertex nomination via attributed random dot product graphs. In Proceedings of the 57th ISI World Statistic Congress.Google Scholar
- Oliveira, R.I. (2010). Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. arXiv:preprint.Google Scholar
- Sussman, D.L. (2014). Foundations of Adjacency Spectral Embedding. PhD Thesis, Johns Hopkins University.Google Scholar
- Tao, T. and Vu, V. (2012). Random matrices: Universal properties of eigenvectors. Random Matrices: Theory and Applications 1.Google Scholar
- Young, S. and Scheinerman, E. (2007). Random dot product graph models for social networks. In Algorithms and models for the web-graph. Springer, p. 138–149.Google Scholar