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

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 es...

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Bibliographic Details
Published in:Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 84; no. 4; pp. 1446 - 1473
Main Authors: Rubin‐Delanchy, Patrick, Cape, Joshua, Tang, Minh, Priebe, Carey E.
Format: Journal Article
Language:English
Published: Oxford Oxford University Press 01.09.2022
Subjects:
Apexes
Clustering
computer security
Eigenvalues
Embedding
Estimates
graph embedding
Internet
Membership
networks
prediction
Probabilistic models
Regression analysis
Representations
spectral clustering
Statistical methods
Statistics
stochastic block model
Stochastic models
ISSN:1369-7412, 1467-9868
Online Access:Get full text
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Summary: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.
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ISSN:1369-7412
1467-9868
DOI:10.1111/rssb.12509