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A statistical interpretation of spectral embedding: The generalised random dot product graph

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  • Patrick Rubin‐Delanchy
  • Joshua Cape
  • Minh Tang
  • Carey E. Priebe

Abstract

Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow interpretation of those vector representations as latent position estimates. The generalisation is needed to model heterophilic connectivity (e.g. ‘opposites attract’) and to cope with negative eigenvalues more generally. We show that, whether the adjacency or normalised Laplacian matrix is used, spectral embedding produces uniformly consistent latent position estimates with asymptotically Gaussian error (up to identifiability). The standard and mixed membership stochastic block models are special cases in which the latent positions take only K distinct vector values, representing communities, or live in the (K − 1)‐simplex with those vertices respectively. Under the stochastic block model, our theory suggests spectral clustering using a Gaussian mixture model (rather than K‐means) and, under mixed membership, fitting the minimum volume enclosing simplex, existing recommendations previously only supported under non‐negative‐definite assumptions. Empirical improvements in link prediction (over the random dot product graph), and the potential to uncover richer latent structure (than posited under the standard or mixed membership stochastic block models) are demonstrated in a cyber‐security example.

Suggested Citation

  • 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.
  • Handle: RePEc:bla:jorssb:v:84:y:2022:i:4:p:1446-1473
    DOI: 10.1111/rssb.12509
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    References listed on IDEAS

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    1. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    2. Zhu, Mu & Ghodsi, Ali, 2006. "Automatic dimensionality selection from the scree plot via the use of profile likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 918-930, November.
    3. 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.
    4. Xueyu Mao & Purnamrita Sarkar & Deepayan Chakrabarti, 2021. "Estimating Mixed Memberships With Sharp Eigenvector Deviations," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1928-1940, October.
    5. J Cape & M Tang & C E Priebe, 2019. "Signal-plus-noise matrix models: eigenvector deviations and fluctuations," Biometrika, Biometrika Trust, vol. 106(1), pages 243-250.
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    Cited by:

    1. Robert Lunde & Purnamrita Sarkar, 2023. "Subsampling sparse graphons under minimal assumptions," Biometrika, Biometrika Trust, vol. 110(1), pages 15-32.
    2. Vainora, J., 2024. "Latent Position-Based Modeling of Parameter Heterogeneity," Cambridge Working Papers in Economics 2455, Faculty of Economics, University of Cambridge.

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