The Annals of Applied Probability

Consistency of modularity clustering on random geometric graphs

Erik Davis and Sunder Sethuraman

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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 $\mathcal{X}_{n}=\{X_{1},X_{2},\ldots,X_{n}\}$, distributed according to a probability measure $\nu$ supported on a bounded domain $D\subset\mathbb{R}^{d}$. Among other results, we show, via a Gamma convergence framework, a geometric form of consistency: When the number of clusters, or partitioning sets of $\mathcal{X}_{n}$ 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.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2003-2062.

Received: April 2016
Revised: February 2017
First available in Project Euclid: 9 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 62G20: Asymptotic properties 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30] 49J55: Problems involving randomness [See also 93E20] 49J45: Methods involving semicontinuity and convergence; relaxation 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35]

Modularity community detection consistency random geometric graph Gamma convergence Kelvin’s problem scaling limit shape optimization optimal transport total variation perimeter


Davis, Erik; Sethuraman, Sunder. Consistency of modularity clustering on random geometric graphs. Ann. Appl. Probab. 28 (2018), no. 4, 2003--2062. doi:10.1214/17-AAP1313.

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