# A modularity-maximization-based approach for detecting multi-communities in social networks

- 18 Downloads

## Abstract

The modularity is a widely-used objective function to determine communities from a given network. The leading eigenvector method is a popular solution that applies the first eigenvector to determine the communities. The low computation cost is the major advantage of the leading eigenvector method. However, the leading eigenvector method only can split a network into two communities. To detect multiple communities, the modularity maximization is transformed to the vector partition problem (VPP). We propose an algorithm which is called as the partition at polar coordinate protocol (PPCP) to solve the VPP problem. The goal of PPCP is to find non-overlapping vertex vector sets so as to maximize the quadratic sum of the norms of community vectors. The proposed PPCP has two steps to determine the communities that are the network structure analysis and the community determination. During the network structure analysis, we obtain following issues. First, the vertex vectors belong to different communities can be separated by the distribution angles. Second, a node with a higher degree corresponds to a vertex vector with a larger norm. So, we propose three refinement functions including the noise reduction, the common-friends model and the strong connectivity hypothesis to improve the accuracy of PPCP. In our simulations, PPCP detects communities more precisely than Fine-tuned algorithm especially in the network with the weak structure. Moreover, the proposed refinement functions can capture the special properties of the network. So, PPCP with refinement functions performs much better than Fine-tuned algorithm and PPCP without refinement functions in terms of the accuracy in detecting communities.

## Keywords

Modularity maximization Community detection Spectral clustering## Notes

## Supplementary material

## References

- Agarwal, G., & Kempe, D. (2008). Modularity-maximizing graph communities via mathematical programming.
*The European Physical Journal B*,*66*(3), 409–418.CrossRefGoogle Scholar - Alfalahi, K., Atif, Y., & Harous, S. (2013). Community detection in social networks through similarity virtual networks. In
*IEEE/ACM international conference on advances in social networks analysis and mining (ASONAM)*, 2013, pp. 1116–1123.Google Scholar - Behera, R. K., & Rath, S. Ku. (January 2016). An efficient modularity based algorithm for community detection in social network. In
*International conference on internet of things and applications (IOTA)*, pp. 22–24.Google Scholar - Chen, M., Kuzmin, K., & Szymanski, B. K. (2014). Community detection via maximization of modularity and its variants.
*IEEE Transactions on Computational Social Systems*,*1*(1), 46–65.CrossRefGoogle Scholar - Chen, M., Nguyen, T., & Szymanski, B. K. (2013a). A new metric for quality of network community structure.
*ASE Human Journal*,*2*(4), 226–240.Google Scholar - Chen, M., Nguyen, T., & Szymanski, B. K. (September 2013b). On measuring the quality of a network community structure. In
*Proceedings of ASE/IEEE international conference on social computing, Alexandria*, pp. 122–127, Virginia.Google Scholar - Clauset, A., Newman, M. E. J., & Moore, C. (2004). Finding community structure in very large networks.
*Physics Review E*,*70*, 066111.CrossRefGoogle Scholar - Duch, J., & Arenas, A. (2005). Community detection in complex networks using extremal optimization.
*Physics Review E*,*72*, 027104.CrossRefGoogle Scholar - Fortunato, S., & Barth’elemy, M. (2007). Resolution limit in community detection.
*Proceedings of the National Academy of Sciences*,*104*(1), 36–41.CrossRefGoogle Scholar - Girvan, M., & Newman, M. E. J. (2002). Community structure in social and biological networks.
*Proceedings of the National Academy of Sciences*,*99*(12), 7821–7826.CrossRefGoogle Scholar - Guimera, R., & Amaral, L. A. N. (2005). Functional cartography of complex metabolic networks.
*Nature*,*433*, 895–900.CrossRefGoogle Scholar - Huang, J., Sun, H., Han, J., Deng, H., Sun, Y., & Liu, Y. (2010). SHRINK: a structural clustering algorithm for detecting hierarchical communities in networks. In
*Proceedings of the 19th conference on information and knowledge management*, New York, NY, pp. 219–228.Google Scholar - Huberman, B. A., Romero, D. M., & Wu, F. (2008). Social networks that matter: Twitter under the microscope. arXiv preprint, arXiv:0812.1045.
- Lancichinetti, A., Fortunato, S., & Radicchi, F. (2008). Benchmark graphs for testing community detection algorithms.
*Physics Review E*,*78*, 046110.CrossRefGoogle Scholar - Lu, X., Kuzmin, K., Chen, M., & Szymanski, B. K. (2018). Adaptive modularity maximization via edge weighting scheme.
*Information Sciences*,*424*, 55–68.CrossRefGoogle Scholar - Moosavi, S. A., Jalali, M., Misaghian, N., Shamshirban, S., & Anisi, M. H. (2017). Community detection in social networks using user frequent pattern mining.
*Knowledge and Information Systems*,*51*(1), 159186.CrossRefGoogle Scholar - Newman, M. E. J. (2003). Fast algorithm for detecting community structure in networks.
*Physics Review E*,*69*, 066133.CrossRefGoogle Scholar - Newman, M. E. J. (2006). Finding community structure in networks using the eigenvectors of matrices.
*Physics Review E*,*74*, 036104.CrossRefGoogle Scholar - Newman, M. E. J. (2006). Modularity and community structure in networks.
*Proceedings of the National Academy of Sciences*,*103*(23), 8577–8582.CrossRefGoogle Scholar - Pothen, A., Simon, H. D., & Liou, K. P. (1990). Partitioning sparse matrices with eigenvectors of graphs.
*SIAM Journal on Matrix Analysis and Applications*,*11*(3), 430452.CrossRefGoogle Scholar - Richardson, T., Mucha, P. J., & Porter, M. A. (2009). Spectral tripartitioning of networks.
*Physics Review E*,*80*, 036111.CrossRefGoogle Scholar - Ruan, J., & Zhang, W. (2008). Identifying network communities with a high resolution.
*Physics Review E*,*77*, 016104.CrossRefGoogle Scholar - Topchy, A. P., Law, M. H., Jain, A. K., & Fred, A. L. (November 2004). Analysis of consensus partition in cluster ensemble. In
*IEEE international conference on data mining*, pp. 225–232.Google Scholar - Van Den Elzen, S., Holten, D., Blaas, J., & Van Wijk, J. J. (2016). Reducing snapshots to points: A visual analytics approach to dynamic network exploration.
*IEEE Transactions on Visualization and Computer Graphics*,*22*(1), 1–10.CrossRefGoogle Scholar - Wanger, S., & Wanger, D. (2007). comparing clusterings—An overview. Universitt Karlsruhe, Technical Report 2006-04.Google Scholar
- White, S., & Smyth, P. (2005). A spectral clustering approach to finding communities in graph. In
*SIAM international conference on data mining*(pp. 274–285). California: Newport Beach.Google Scholar - Zachary, W. W. (1977). An information flow model for conflict and fission in small groups.
*Journal of Anthropological Research*,*33*, 452–473.CrossRefGoogle Scholar