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Regularized spectral methods for clustering signed networks

Mihai Cucuringu 1 Apoorv Vikram Singh 2 Deborah Sulem 1 Hemant Tyagi 3
3 MODAL - MOdel for Data Analysis and Learning
Inria Lille - Nord Europe, LPP - Laboratoire Paul Painlevé - UMR 8524, METRICS - Evaluation des technologies de santé et des pratiques médicales - ULR 2694, Polytech Lille - École polytechnique universitaire de Lille, Université de Lille, Sciences et Technologies
Abstract : We study the problem of k-way clustering in signed graphs. Considerable attention in recent years has been devoted to analyzing and modeling signed graphs, where the affinity measure betweennodes takes either positive or negative values. Recently, [CDGT19] proposed a spectral method,namely SPONGE (Signed Positive over Negative Generalized Eigenproblem), which casts the clustering task as a generalized eigenvalue problem optimizing a suitably defined objective function.This approach is motivated by social balance theory, where the clustering task aims to decomposea given network into disjoint groups, such that individuals within the same group are connected byas many positive edges as possible, while individuals from different groups are mainly connected bynegative edges. Through extensive numerical simulations, SPONGE was shown to achieve state-of-the-art empirical performance. On the theoretical front, [CDGT19] analyzed SPONGE, as well asthe popular Signed Laplacian based spectral method under the setting of a Signed Stochastic BlockModel, for k=2 equal-sized clusters, in the regime where the graph is moderately dense. In this work, we build on the results in [CDGT19] on two fronts for the normalized versions of SPONGE and the Signed Laplacian. Firstly, for both algorithms, we extend the theoretical analysisin [CDGT19] to the general setting of k>2 unequal-sized clusters in the moderately dense regime. Secondly, we introduce regularized versions of both methods to handle sparse graphs – a regime where standard spectral methods are known to underperform – and provide theoretical guaranteesunder the same setting of a Signed Stochastic Block Model. To the best of our knowledge, regularized spectral methods have so far not been considered in the setting of clustering signed graphs. Wecomplement our theoretical results with an extensive set of numerical experiments on synthetic data.
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Submitted on : Thursday, January 7, 2021 - 11:36:16 AM
Last modification on : Friday, January 21, 2022 - 3:09:35 AM
Long-term archiving on: : Thursday, April 8, 2021 - 6:50:14 PM

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Mihai Cucuringu, Apoorv Vikram Singh, Deborah Sulem, Hemant Tyagi. Regularized spectral methods for clustering signed networks. Journal of Machine Learning Research, Microtome Publishing, 2021, 22 (264), pp.1-79. ⟨hal-03101710⟩

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