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.
Bibliographical noteFunding 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.
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.
© 2018 Society for Industrial and Applied Mathematics.
- Community detection
- Completely positive relaxation
- Degree-correlated stochastic block model
- Nonasymptotic error bound
- Proximal block coordinate descent method