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Modularity functions maximization with nonnegative relaxation facilitates community detection in networks

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  • Jiang, Jonathan Q.
  • McQuay, Lisa J.

Abstract

We show here that the problem of maximizing a family of quantitative functions, encompassing both the modularity (Q-measure) and modularity density (D-measure), for community detection can be uniformly understood as a combinatoric optimization involving the trace of a matrix called modularity Laplacian. Instead of using traditional spectral relaxation, we apply additional nonnegative constraint into this graph clustering problem and design efficient algorithms to optimize the new objective. With the explicit nonnegative constraint, our solutions are very close to the ideal community indicator matrix and can directly assign nodes into communities. The near-orthogonal columns of the solution can be reformulated as the posterior probability of corresponding node belonging to each community. Therefore, the proposed method can be exploited to identify the fuzzy or overlapping communities and thus facilitates the understanding of the intrinsic structure of networks. Experimental results show that our new algorithm consistently, sometimes significantly, outperforms the traditional spectral relaxation approaches.

Suggested Citation

  • Jiang, Jonathan Q. & McQuay, Lisa J., 2012. "Modularity functions maximization with nonnegative relaxation facilitates community detection in networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 854-865.
  • Handle: RePEc:eee:phsmap:v:391:y:2012:i:3:p:854-865
    DOI: 10.1016/j.physa.2011.08.043
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    References listed on IDEAS

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    1. Gergely Palla & Imre Derényi & Illés Farkas & Tamás Vicsek, 2005. "Uncovering the overlapping community structure of complex networks in nature and society," Nature, Nature, vol. 435(7043), pages 814-818, June.
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    Cited by:

    1. Shang, Ronghua & Bai, Jing & Jiao, Licheng & Jin, Chao, 2013. "Community detection based on modularity and an improved genetic algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(5), pages 1215-1231.
    2. Santiago, Rafael & Lamb, Luís C., 2017. "Efficient modularity density heuristics for large graphs," European Journal of Operational Research, Elsevier, vol. 258(3), pages 844-865.
    3. Shang, Ronghua & Luo, Shuang & Zhang, Weitong & Stolkin, Rustam & Jiao, Licheng, 2016. "A multiobjective evolutionary algorithm to find community structures based on affinity propagation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 203-227.

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