Consistency of modularity clustering on random geometric graphs

Erik Davis, Sunder Sethuraman

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

Given a graph, the popular “modularity” clustering method specifies a partition of the vertex set as the solution of a certain optimization problem. In this paper, we discuss scaling limits of this method with respect to random geometric graphs constructed from i.i.d. points Xn = {X1,X2,...,Xn}, distributed according to a probability measure ν supported on a bounded domain D ⊂ Rd. Among other results, we show, via a Gamma convergence framework, a geometric form of consistency: When the number of clusters, or partitioning sets of Xn is a priori bounded above, the discrete optimal modularity clusterings converge in a specific sense to a continuum partition of the underlying domain D, characterized as the solution to a “soap bubble” or “Kelvin”-type shape optimization problem.

Original languageEnglish (US)
Pages (from-to)2003-2062
Number of pages60
JournalAnnals of Applied Probability
Volume28
Issue number4
DOIs
StatePublished - Aug 1 2018

Keywords

  • Community detection
  • Consistency
  • Gamma convergence
  • Kelvin’s problem
  • Modularity
  • Optimal transport
  • Perimeter
  • Random geometric graph
  • Scaling limit
  • Shape optimization
  • Total variation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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