Consistency of modularity clustering on random geometric graphs

Erik Davis, Sunder Sethuraman

Research output: Contribution to journalArticle

2 Citations (Scopus)

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

Fingerprint

Random Geometric Graph
Modularity
Partition
Clustering
Optimization Problem
Set Partitioning
Gamma Convergence
Scaling Limit
Kelvin
Shape Optimization
Number of Clusters
Clustering Methods
Bubble
Probability Measure
Bounded Domain
Continuum
Converge
Graph in graph theory
Vertex of a graph
Graph

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

Cite this

Consistency of modularity clustering on random geometric graphs. / Davis, Erik; Sethuraman, Sunder.

In: Annals of Applied Probability, Vol. 28, No. 4, 01.08.2018, p. 2003-2062.

Research output: Contribution to journalArticle

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