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Affiliation
Univ Arizona, Dept MathIssue Date
2018-08Keywords
Modularitycommunity detection
consistency
random geometric graph
Gamma convergence
Kelvin's problem
scaling limit
shape optimization
optimal transport
total variation
perimeter
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INST MATHEMATICAL STATISTICSCitation
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. https://projecteuclid.org/euclid.aoap/1533780266Journal
ANNALS OF APPLIED PROBABILITYRights
© Institute of Mathematical Statistics, 2018.Collection Information
This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at repository@u.library.arizona.edu.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 X-n = {X-1, X-2,..., X-n}, distributed according to a probability measure nu supported on a bounded domain D subset of 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 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.ISSN
1050-5164Version
Final published versionSponsors
ARO [W911NF-14-1-0179]Additional Links
https://projecteuclid.org/euclid.aoap/1533780266ae974a485f413a2113503eed53cd6c53
10.1214/17-AAP1313