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    Consistency of modularity clustering on random geometric graphs

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    Author
    Davis, Erik
    Sethuraman, Sunder
    Affiliation
    Univ Arizona, Dept Math
    Issue Date
    2018-08
    Keywords
    Modularity
    community detection
    consistency
    random geometric graph
    Gamma convergence
    Kelvin's problem
    scaling limit
    shape optimization
    optimal transport
    total variation
    perimeter
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    Publisher
    INST MATHEMATICAL STATISTICS
    Citation
    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/1533780266
    Journal
    ANNALS OF APPLIED PROBABILITY
    Rights
    © 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
    1050-5164
    DOI
    10.1214/17-AAP1313
    Version
    Final published version
    Sponsors
    ARO [W911NF-14-1-0179]
    Additional Links
    https://projecteuclid.org/euclid.aoap/1533780266
    ae974a485f413a2113503eed53cd6c53
    10.1214/17-AAP1313
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    UA Faculty Publications

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