### 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 ν supported on a bounded domain D ⊂ 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.

Original language | English (US) |
---|---|

Pages (from-to) | 2003-2062 |

Number of pages | 60 |

Journal | Annals of Applied Probability |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2018 |

### Fingerprint

### 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

*Annals of Applied Probability*,

*28*(4), 2003-2062. https://doi.org/10.1214/17-AAP1313

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

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 28, no. 4, pp. 2003-2062. https://doi.org/10.1214/17-AAP1313

}

TY - JOUR

T1 - Consistency of modularity clustering on random geometric graphs

AU - Davis, Erik

AU - Sethuraman, Sunder

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Community detection

KW - Consistency

KW - Gamma convergence

KW - Kelvin’s problem

KW - Modularity

KW - Optimal transport

KW - Perimeter

KW - Random geometric graph

KW - Scaling limit

KW - Shape optimization

KW - Total variation

UR - http://www.scopus.com/inward/record.url?scp=85052705196&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052705196&partnerID=8YFLogxK

U2 - 10.1214/17-AAP1313

DO - 10.1214/17-AAP1313

M3 - Article

VL - 28

SP - 2003

EP - 2062

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 4

ER -