Skip to main content
Springer Nature Link
Log in

Community detection based on strong Nash stable graph partition

  • Original Article
  • Published:
Social Network Analysis and Mining Aims and scope Submit manuscript

Abstract

The modern science of network mining has brought significant advances to the understanding of complex social networks by studying the modular structures of the networks. The problem of identifying the network modules, also called communities, is an active area of research across several disciplines due to its potential value in real-life applications. The mainstream approach for community detection generally focuses on optimization of a global metric that measures the quality of a partition over a given network. Optimizing a global metric is akin to community assignment by a centralized decision maker. However, communities in a social network often evolve in a decentralized fashion. To model the natural formation of a community, we propose a game theory-based framework treating each node as a player of a hedonic coalition formation game. We propose a novel preference relation for the players based on the players’ preference to form stable and dense community structures. The notion of strong Nash stability is used, for the first time, to determine the outcome of the coalition formation game in the context of community detection. Subsequently, we propose a greedy algorithm to obtain the strong Nash stable partition of the game. Evaluation on the well-known synthetic benchmarks and real-world networks demonstrate the superiority of the proposed algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Abello J, Resende M, Sudarsky S (2002) Massive quasi-clique detection. In: Rajsbaum S (ed) LATIN 2002: theoretical informatics, vol 2286., Lecture notes in computer science. Springer, Berlin, pp 598–612

  • Aloise D, Cafieri S, Caporossi G, Hansen P, Perron S, Liberti L (2010) Column generation algorithms for exact modularity maximization in networks. Phys Rev E 82:046112

    Article  Google Scholar 

  • Altaf-Ul-Amin M, Nishikata K, Koma T, Miyasato T, Shinbo Y, Arifuzzaman M, Wada C, Maeda M, Oshima T, Mori H et al (2003) Prediction of protein functions based on k-cores of protein-protein interaction networks and amino acid sequences. Genome Inf Ser 14:498–499

  • Alvarez-Hamelin JI, Dall’Asta L, Barrat A, Vespignani A (2005) Large scale networks fingerprinting and visualization using the k-core decomposition. In: Advances in neural information processing systems, Vancouver, British Columbia, Canada, pp 41–50

  • Alvari H, Hashemi S, Hamzeh A (2013) Discovering overlapping communities in social networks: a novel game-theoretic approach. AI Commun 26(2):161–177

    MathSciNet  MATH  Google Scholar 

  • Ana L, Jain A (2003) Robust data clustering. In: Proceedings. IEEE computer society conference on computer vision and pattern recognition, vol 2, pp II–128–II–133 vol 2

  • Apicella CL, Marlowe FW, Fowler JH, Christakis NA (2012) Social networks and cooperation in hunter-gatherers. Nature 481(7382):497–501

    Article  Google Scholar 

  • Aziz H, Brandl F (2012) Existence of stability in hedonic coalition formation games. In: Proceedings of the 11th international conference on autonomous agents and multiagent systems, Vol 2, pp 763–770

  • Aziz H, Brandt F, Seedig HG (2011) Stable partitions in additively separable hedonic games. In: The 10th international conference on autonomous agents and multiagent systems, Vol 1, pp 183–190

  • Bader GD, Hogue CW (2003) An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinf 4(1):2

    Article  Google Scholar 

  • Badham J, Stocker R (2010) The impact of network clustering and assortativity on epidemic behaviour. Theor Popul Biol 77(1):71–75

    Article  Google Scholar 

  • Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 2008(10):P10008

    Article  Google Scholar 

  • Bogomolnaia A, Jackson MO (2002) The stability of hedonic coalition structures. Games Econ Behav 38(2):201–230

    Article  MathSciNet  MATH  Google Scholar 

  • Brunato M, Hoos H, Battiti R (2008) On effectively finding maximal quasi-cliques in graphs. Learning and intelligent optimization, vol 5313. Lecture notes in computer science. Springer, Berlin, pp 41–55

  • Chan P, Schlag M, Zien J (1994) Spectral k-way ratio-cut partitioning and clustering. Comput Aided Des Integr Circuits Syst IEEE Trans on 13(9):1088–1096

    Article  Google Scholar 

  • Chen W, Dress A, Yu W (2008) Community structures of networks. Math Comput Sci 1(3):441–457

    Article  MathSciNet  MATH  Google Scholar 

  • Chen W, Liu Z, Sun X, Wang Y (2010) A game-theoretic framework to identify overlapping communities in social networks. Data Mining Knowl Discov 21(2):224–240

    Article  MathSciNet  Google Scholar 

  • Ciglan M, Laclavík M, Nørvåg K (2013) On community detection in real-world networks and the importance of degree assortativity. In: Proceedings of the 19th ACM SIGKDD international conference on knowledge discovery and data mining, KDD ’13, pp 1007–1015

  • Clauset A, Newman MEJ, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70:066111

    Article  Google Scholar 

  • Danon L, Daz-Guilera A, Duch J, Arenas A (2005) Comparing community structure identification. J Stat Mech Theory Exp 2005(09):P09,008

    Article  Google Scholar 

  • Ding C, He X, Zha H, Gu M, Simon H (2001) A min-max cut algorithm for graph partitioning and data clustering. In: Data mining, 2001. ICDM 2001, Proceedings IEEE international conference on, pp 107–114

  • Duch J, Arenas A (2005) Community detection in complex networks using extremal optimization. Phys Rev E 72:027,104

    Article  Google Scholar 

  • Erdős P, Rényi A (1961) On the strength of connectedness of a random graph. Acta Math Hung 12(1):261–267

    Google Scholar 

  • Faulkner J, Schaller M, Park JH, Duncan LA (2004) Evolved disease-avoidance mechanisms and contemporary xenophobic attitudes. Group Process Intergr Relat 7(4):333–353

    Article  Google Scholar 

  • Fincher CL, Thornhill R (2008) Assortative sociality, limited dispersal, infectious disease and the genesis of the global pattern of religion diversity. Proc R Soc Lond B Biol Sci 275(1651):2587–2594

    Article  Google Scholar 

  • Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174

    Article  MathSciNet  Google Scholar 

  • Fortunato S, Barthélemy M (2007) Resolution limit in community detection. Proc Natl Acad Sci 104(1):36–41

    Article  Google Scholar 

  • Gatti N, Rocco M, Sandholm T (2013) On the complexity of strong nash equilibrium: hard-to-solve instances and smoothed complexity. arXiv preprint arXiv:1304.1351

  • Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci 99(12):7821–7826

    Article  MathSciNet  MATH  Google Scholar 

  • Guimerà R, Sales-Pardo M, Amaral LAN (2004) Modularity from fluctuations in random graphs and complex networks. Phys Rev E 70:025,101

    Article  Google Scholar 

  • Gunes I, Bingol H (2006) Community detection in complex networks using agents. arXiv preprint cs/0610129

  • Heimo T, Kumpula JM, Kaski K, Saramki J (2008) Detecting modules in dense weighted networks with the potts method. J Stat Mech Theory Exp 2008(08):P08,007

    Article  Google Scholar 

  • Karakaya M (2011) Hedonic coalition formation games: a new stability notion. Math Soc Sci 61(3):157–165

    Article  MathSciNet  MATH  Google Scholar 

  • Kernighan BW, Lin S (1970) An efficient heuristic procedure for partitioning graphs. Bell Syst Tech J 49(2):291–307

    Article  MATH  Google Scholar 

  • Krishnamurthy B, Wang J (2000) On network-aware clustering of web clients. SIGCOMM Comput Commun Rev 30(4):97–110

    Article  Google Scholar 

  • Lancichinetti A, Fortunato S, Radicchi F (2008) Benchmark graphs for testing community detection algorithms. Phys Rev E 78:046,110

    Article  Google Scholar 

  • Lau HC, Zhang L (2003) Task allocation via multi-agent coalition formation: taxonomy, algorithms and complexity. In: Tools with artificial intelligence, 2003. Proceedings of 15th IEEE international conference on, pp 346–350

  • Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data 1(1)

  • Leskovec J, Lang KJ, Dasgupta A, Mahoney MW (2009) Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters. Internet Math 6(1):29–123

    Article  MathSciNet  MATH  Google Scholar 

  • Leskovec J, Lang KJ, Mahoney M (2010) Empirical comparison of algorithms for network community detection. In: Proceedings of the 19th international conference on world wide web, pp 631–640

  • Mahdavi Pajouh F, Miao Z, Balasundaram B (2014) A branch-and-bound approach for maximum quasi-cliques. Ann Oper Res 216(1):145–161

    Article  MathSciNet  Google Scholar 

  • Martin SB, Brown WM, Klavans R, Boyack KW (2008) Drl: distributed recursive (graph) layout. Tech rep, Sandia National Laboratories (SNL-NM), Albuquerque, NM, USA

  • Mastrandrea R, Fournet J, Barrat A (2015) Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. arXiv:1506.03645

  • Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298(5594):824–827

    Article  Google Scholar 

  • Molnár Jr F, Derzsy N, Szymanski B, Korniss G (2014) Building damage-resilient dominating sets in complex networks against random and targeted attacks. CoRR. http://arxiv.org/abs/1409.7831

  • Moody J (2001) Peer influence groups: identifying dense clusters in large networks. Soc Netw 23(4):261–283

    Article  Google Scholar 

  • Narayanam R, Narahari Y (2012) A game theory inspired, decentralized, local information based algorithm for community detection in social graphs. In: 21st international conference on pattern recognition (ICPR), pp 1072–1075

  • Navarrete CD, Fessler DM (2006) Disease avoidance and ethnocentrism: the effects of disease vulnerability and disgust sensitivity on intergroup attitudes. Evol Hum Behav 27(4):270–282

    Article  Google Scholar 

  • Newman M (2003) Ego-centered networks and the ripple effect. Soc Netw 25(1):83–95

    Article  Google Scholar 

  • Newman ME (2002) Random graphs as models of networks. arXiv preprint cond-mat/0202208

  • Newman MEJ (2004) Fast algorithm for detecting community structure in networks. Phys Rev E 69:066,133

    Article  Google Scholar 

  • Newman MEJ (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036–104

    Article  Google Scholar 

  • Newman MEJ (2006) Modularity and community structure in networks. Proc Natl Acad Sci 103(23):8577–8582

    Article  Google Scholar 

  • Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026,113

    Article  Google Scholar 

  • Pattillo J, Youssef N, Butenko S (2013) Clique relaxation models in social network analysis. Eur J Oper Res 226(1):9–18

    Article  MathSciNet  MATH  Google Scholar 

  • Pechenick DA, Payne JL, Moore JH (2012) The influence of assortativity on the robustness of signal-integration logic in gene regulatory networks. J Theor Biol 296:21–32

    Article  Google Scholar 

  • Rahwan T, Jennings NR (2008) Coalition structure generation: dynamic programming meets anytime optimization. AAAI 8:156–161

    Google Scholar 

  • Sandholm T, Larson K, Andersson M, Shehory O, Tohm F (1999) Coalition structure generation with worst case guarantees. Artif Intell 111(1–2):209–238

    Article  MATH  Google Scholar 

  • Shehory O, Kraus S (1998) Methods for task allocation via agent coalition formation. Artif Intell 101(1–2):165–200

    Article  MathSciNet  MATH  Google Scholar 

  • Shi J, Malik J (2000) Normalized cuts and image segmentation. Pattern Anal Mach Intell IEEE Trans On 22(8):888–905

    Article  Google Scholar 

  • Tsourakakis C, Bonchi F, Gionis A, Gullo F, Tsiarli M (2013) Denser than the densest subgraph: Extracting optimal quasi-cliques with quality guarantees. In: Proceedings of the 19th ACM SIGKDD international conference on knowledge discovery and data mining, pp 104–112

  • Uno T (2007) An efficient algorithm for enumerating pseudo cliques. Algorithms and computation, vol 4835. Lecture notes in computer science. Springer, Berlin, pp 402–414

  • Yang J, Leskovec J (2015) Defining and evaluating network communities based on ground-truth. Knowl Inf Syst 42(1):181–213

    Article  Google Scholar 

  • Zachary WW (1977) An information flow model for conflict and fission in small groups. J Anthropol Res 33(4):452–473

    Google Scholar 

Download references

Acknowledgments

The authors express their sincere gratitude to the anonymous reviewers for their valuable suggestions. The suggestions and the comments helped the authors to improve the quality of the manuscript.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Jadavpur University, Kolkata, India

    Srinka Basu & Ujjwal Maulik

Authors
  1. Srinka Basu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Ujjwal Maulik
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Srinka Basu.

Appendix

Appendix

As modularity (Q) (Newman and Girvan 2004) is a popular goodness metric, its values might be of interest to the readers. However, we exclude the metric from the experiment as otherwise that would make the experiment biased toward the modularity optimization methods like LM. Here, in Table 6 we provide the values of Q for some of the data sets used in the experiment. We also include the modularity of the ground truth partitions where applicable. As shown in the results, LM, which directly optimizes Q, attains maximum Q in all the data sets. It is particularly important to note that the modularity attained by LM is even higher than that of the ground truth partitions, indicating that high modularity does not necessarily indicate a meaningful partition.

Table 6 Modularity values of the partitions
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Basu, S., Maulik, U. Community detection based on strong Nash stable graph partition. Soc. Netw. Anal. Min. 5, 61 (2015). https://doi.org/10.1007/s13278-015-0299-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13278-015-0299-4

Keywords