Open Access
2022 Spectral graph clustering via the expectation-solution algorithm
Zachary M. Pisano, Joshua S. Agterberg, Carey E. Priebe, Daniel Q. Naiman
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Electron. J. Statist. 16(1): 3135-3175 (2022). DOI: 10.1214/22-EJS2018


The stochastic blockmodel (SBM) models the connectivity within and between disjoint subsets of nodes in networks. Prior work demonstrated that the rows of an SBM’s adjacency spectral embedding (ASE) and Laplacian spectral embedding (LSE) both converge in law to Gaussian mixtures where the components are curved exponential families. Maximum likelihood estimation via the Expectation-Maximization (EM) algorithm for a full Gaussian mixture model (GMM) can then perform the task of clustering graph nodes, albeit without appealing to the components’ curvature. Noting that EM is a special case of the Expectation-Solution (ES) algorithm, we propose two ES algorithms that allow us to take full advantage of these curved structures. After presenting the ES algorithm for the general curved-Gaussian mixture, we develop those corresponding to the ASE and LSE limiting distributions. Simulating from artificial SBMs and a brain connectome SBM reveals that clustering graph nodes via our ES algorithms can improve upon that of EM for a full GMM for a wide range of settings.

Funding Statement

This work is partially supported by the D3M program of the Defense Advanced Research Projects Agency (DARPA). The authors would like to acknowledge and thank the referees, whose commentary and criticism of an earlier version of the present article improved it considerably.


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Zachary M. Pisano. Joshua S. Agterberg. Carey E. Priebe. Daniel Q. Naiman. "Spectral graph clustering via the expectation-solution algorithm." Electron. J. Statist. 16 (1) 3135 - 3175, 2022.


Received: 1 January 2021; Published: 2022
First available in Project Euclid: 11 May 2022

MathSciNet: MR4419471
zbMATH: 1493.62397
Digital Object Identifier: 10.1214/22-EJS2018

Primary: 60B20 , 62H30
Secondary: 62P10

Keywords: curved exponential family , EM algorithm , estimating equations , mixture model , random graph

Vol.16 • No. 1 • 2022
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