A sparse completely positive relaxation of the modularity maximization for community detection

Junyu Zhang; Haoyang Liu; Zaiwen Wen; Shuzhong Zhang

doi:10.1137/17M1141904

A sparse completely positive relaxation of the modularity maximization for community detection

Junyu Zhang, Haoyang Liu, Zaiwen Wen, Shuzhong Zhang

Industrial and Systems Engineering

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated SBM assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR)-type block coordinate descent algorithm to solve the relaxation and prove an O(1/√N) convergence rate to a stationary point, where N is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from a solution to the above relaxation. Nonasymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a competitive alternative for the community detection problem on sparse networks with over 50 million nodes.

Original language	English (US)
Pages (from-to)	A3091-A3200
Journal	SIAM Journal on Scientific Computing
Volume	40
Issue number	5
DOIs	https://doi.org/10.1137/17M1141904
State	Published - 2018

Bibliographical note

Funding Information:
\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section August 3, 2017; accepted for publication (in revised form) July 9, 2018; published electronically September 25, 2018. http://www.siam.org/journals/sisc/40-5/M114190.html \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The first and the fourth authors' research was supported in part by the National Science Foundation under grants CMMI-1462408 and FAIN-1723529. The third author's work was supported in part by the NSFC grants 11831002 and 11421101, and by the National Basic Research Project under the grant 2015CB856002.

Funding Information:
The first and the fourth authors' research was supported in part by the National Science Foundation under grants CMMI-1462408 and FAIN-1723529. The third author's work was supported in part by the NSFC grants 11831002 and 11421101, and by the National Basic Research Project under the grant 2015CB856002.

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

Keywords

Community detection
Completely positive relaxation
Degree-correlated stochastic block model
Nonasymptotic error bound
Proximal block coordinate descent method

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title = "A sparse completely positive relaxation of the modularity maximization for community detection",

abstract = "In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated SBM assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR)-type block coordinate descent algorithm to solve the relaxation and prove an O(1/√N) convergence rate to a stationary point, where N is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from a solution to the above relaxation. Nonasymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a competitive alternative for the community detection problem on sparse networks with over 50 million nodes.",

keywords = "Community detection, Completely positive relaxation, Degree-correlated stochastic block model, Nonasymptotic error bound, Proximal block coordinate descent method",

author = "Junyu Zhang and Haoyang Liu and Zaiwen Wen and Shuzhong Zhang",

note = "Funding Information: \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section August 3, 2017; accepted for publication (in revised form) July 9, 2018; published electronically September 25, 2018. http://www.siam.org/journals/sisc/40-5/M114190.html \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The first and the fourth authors' research was supported in part by the National Science Foundation under grants CMMI-1462408 and FAIN-1723529. The third author's work was supported in part by the NSFC grants 11831002 and 11421101, and by the National Basic Research Project under the grant 2015CB856002. Funding Information: The first and the fourth authors' research was supported in part by the National Science Foundation under grants CMMI-1462408 and FAIN-1723529. The third author's work was supported in part by the NSFC grants 11831002 and 11421101, and by the National Basic Research Project under the grant 2015CB856002. Publisher Copyright: {\textcopyright} 2018 Society for Industrial and Applied Mathematics.",

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N2 - In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated SBM assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR)-type block coordinate descent algorithm to solve the relaxation and prove an O(1/√N) convergence rate to a stationary point, where N is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from a solution to the above relaxation. Nonasymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a competitive alternative for the community detection problem on sparse networks with over 50 million nodes.

AB - In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated SBM assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR)-type block coordinate descent algorithm to solve the relaxation and prove an O(1/√N) convergence rate to a stationary point, where N is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from a solution to the above relaxation. Nonasymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a competitive alternative for the community detection problem on sparse networks with over 50 million nodes.

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KW - Degree-correlated stochastic block model

KW - Nonasymptotic error bound

KW - Proximal block coordinate descent method

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A sparse completely positive relaxation of the modularity maximization for community detection

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Keywords

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