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August 2018 Consistency of modularity clustering on random geometric graphs
Erik Davis, Sunder Sethuraman
Ann. Appl. Probab. 28(4): 2003-2062 (August 2018). DOI: 10.1214/17-AAP1313

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 $\mathcal{X}_{n}=\{X_{1},X_{2},\ldots,X_{n}\}$, 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.

Citation

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Erik Davis. Sunder Sethuraman. "Consistency of modularity clustering on random geometric graphs." Ann. Appl. Probab. 28 (4) 2003 - 2062, August 2018. https://doi.org/10.1214/17-AAP1313

Information

Received: 1 April 2016; Revised: 1 February 2017; Published: August 2018
First available in Project Euclid: 9 August 2018

zbMATH: 06974744
MathSciNet: MR3843822
Digital Object Identifier: 10.1214/17-AAP1313

Subjects:
Primary: 60D05
Secondary: 05C82, 49J45, 49J55, 62G20, 68R10

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 4 • August 2018
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