Skip to main content
SpringerLink
Log in

Determining maximum cliques for community detection in weighted sparse networks

  • Regular Paper
  • Published:
Knowledge and Information Systems Aims and scope Submit manuscript

Abstract

In this article, we have delved into detecting dense regions of weighted sparse networks through identification of cliques and communities. A novel clique finding method is introduced to generate all maximum cliques of a sparse network, with focus on analysis of real-life networks and a community detection algorithm is devised on maximal cliques to determine possible overlapping communities of the weighted sparse network. A good number of methods of clique detection already exist, some of which are truly efficient, but many of them lack direct applicability to real-life network analysis, as they deal with simple networks, hide intermediary details and allow cliques to be formed without information on strength of interactions in group behavior. The proposed method attaches a clique-intensity value with every clique and using two thresholds on intensity of interactions, at the individual level and group level, provides handle to filter stray or insignificant interactions at various stages of clique formation. Using differently sized weighted maximal cliques as building blocks, a new overlapping community detection method is proposed using a new measure, a weighted version of Jaccard index, called weighted Jaccard index. Experiments are done on real-life networks; the maximum clique structures reveal sensitivity to threshold values, while the resulting community structures demonstrate efficacy of the community detection method.

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

Access this article

Log in via an institution

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
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Abello J, Pardalos PM, Resende MGC (1998) On very large maximum clique problems. In: Proceedings of algorithms and experiments (ALEX98), pp 175–183

  2. Aharony N, Pan W, Ip C, Khayal I, Pentland A (2011) Social fMRI: investigating and shaping social mechanisms in the real world. Pervasive Mobile Comput 7(6):643–659

    Article  Google Scholar 

  3. Alidaee B, Glover F, Kochenberger G, Wang H (2007) Solving the maximum edge weight clique problem via unconstrained quadratic programming. Eur J Oper Res 181(2):592–597

    Article  MATH  Google Scholar 

  4. Bahadur KC, Akutsu T, Tomita E, Seki T (2004) Protein side-chain packing problem: a maximum edge-weight clique algorithmic approach. In: Proceedings of the second conference on asia-pacific bioinformatics-volume 29. Australian Computer Society, Inc., pp 191–200

  5. Barabási AL (2016) Network science. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  6. Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A (2004) The architecture of complex weighted networks. Proc Natl Acad Sci 101(11):3747–3752

    Article  MATH  Google Scholar 

  7. Batsyn M, Goldengorin B, Maslov E, Pardalos PM (2014) Improvements to MCS algorithm for the maximum clique problem J. Combin Optim 27(2):397–416

    Article  MathSciNet  MATH  Google Scholar 

  8. Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem D. In: Du, Pardalos PM (eds) Handbook of combinatorial optimization. Springer, New York, pp 1–74

    Google Scholar 

  9. Cai S, Su K, Sattar A (2011) Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artif Intell 175(9–10):1672–1696

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen D, Shang M, Lv Z, Fu Y (2010) Detecting overlapping communities of weighted networks via a local algorithm. Physica A: Stat Mech Appl 389(19):4177–4187

    Article  Google Scholar 

  11. Chu Y, Liu B, Cai S, Luo C, You H (2020) An efficient local search algorithm for solving maximum edge weight clique problem in large graphs. J Combin Optim 34:933–954

    Article  MathSciNet  MATH  Google Scholar 

  12. Clauset A, Newman ME, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70(6):066111

    Article  Google Scholar 

  13. Das A, Svendsen M, Tirthapura S (2019) Incremental maintenance of maximal cliques in a dynamic graph. VLDB J 28(3):351–375

    Article  Google Scholar 

  14. Diestel R (2000) Graph theory {Graduate Texts in Mathematics; 173}. Springer, Berlin

    Google Scholar 

  15. Eppstein D, Löffler M, Strash D (2010) December. Listing all maximal cliques in sparse graphs in near-optimal time. In: International symposium on algorithms and computation. Springer, Berlin, pp 403–414

  16. Eppstein D, Strash D (2011) Listing all maximal cliques in large sparse real-world graphs. Exp Algorithms 18:364–375

    Article  MATH  Google Scholar 

  17. Friden C, Hertz A, de Werra D (1989) Stabulus: a technique for finding stable sets in large graphs with tabu search. Computing 42(1):35–44

    Article  MATH  Google Scholar 

  18. Gong Y, Zhu Y, Duan L, Liu Q, Guan Z, Sun F, Ou W, Zhu KQ (2019) Exact-k recommendation via maximal clique optimization. In: Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery and data mining, pp 617–626

  19. Gouveia L, Pedro M (2015) Solving the maximum edge-weight clique problem in sparse graphs with compact formulations. EURO J Comput Optim 3(1):1–30

    Article  MathSciNet  MATH  Google Scholar 

  20. Hosseinian S, Fontes DB, Butenko S, Nardelli MB, Fornari M, Curtarolo S (2017) The maximum edge weight clique problem: formulations and solution approaches. In: Optimization methods and applications. Springer, Cham, pp 217–237

  21. Jaccard P (1902) Distribution comparée de la flore alpine dans quelques régions des Alpes occidentales et orientales. Bulletin de la Murithienne 31:81–92

    Google Scholar 

  22. Jiang H, Li CM, Manya F (2017) An exact algorithm for the maximum weight clique problem in large graphs. In: Thirty-first AAAI conference on artificial intelligence.

  23. Konc J, Janezic D (2007) An improved branch and bound algorithm for the maximum clique problem. Proteins 4:5

    MATH  Google Scholar 

  24. Knuth DE (1993) The Stanford graphbase: a platform for combinatorial computing. ACM Press, New York, pp 74–87

    MATH  Google Scholar 

  25. Lu Z, Wahlström J, Nehorai A (2018) Community detection in complex networks via clique conductance. Sci Rep 8(1):1–16

    Google Scholar 

  26. Malladi KT, Mitrovic-Minic S, Punnen AP (2017) Clustered maximum weight clique problem: algorithms and empirical analysis. Comput Oper Res 85:113–128

    Article  MathSciNet  MATH  Google Scholar 

  27. Martí R, Gallego M, Duarte A (2010) A branch and bound algorithm for the maximum diversity problem. Eur J Oper Res 200(1):36–44

    Article  MATH  Google Scholar 

  28. Martins P (2010) Extended and discretized formulations for the maximum clique problem. Comput Oper Res 37(7):1348–1358

    Article  MathSciNet  MATH  Google Scholar 

  29. Mehrotra A, Trick MA (1998) Cliques and clustering: A combinatorial approach. Oper Res Lett 22(1):1–12

    Article  MathSciNet  MATH  Google Scholar 

  30. Nešetřil J, de Mendez PO (2012) Sparsity: graphs, structures, and algorithms, vol 28. Springer, Berlin

    MATH  Google Scholar 

  31. Newman ME (2001) Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys Rev E 64(1):016132

    Article  Google Scholar 

  32. Newman ME (2003) The structure and function of complex networks. SIAM Rev 45(2):167–256

    Article  MathSciNet  MATH  Google Scholar 

  33. Newman ME (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036104

    Article  MathSciNet  Google Scholar 

  34. Östergård PR (2001) A new algorithm for the maximum-weight clique problem. Nordic J Comput 8(4):424–436

    MathSciNet  MATH  Google Scholar 

  35. Palla G, Ábel D, Derényi I, Farkas I, Pollner P, Vicsek T (2005) K-clique percolation and clustering in directed and weighted networks. Bolayai Society Mathematical Studies, Berlin

    Google Scholar 

  36. Palla G, Derényi I, Farkas I, Vicsek T (2005) Uncovering the overlapping community structure of complex networks in nature and society. Nature 435:814–818

    Article  Google Scholar 

  37. Park K, Lee K, Park S (1996) An extended formulation approach to the edge-weighted maximal clique problem. Eur J Oper Res 95(3):671–682

    Article  MATH  Google Scholar 

  38. Richter S, Helmert M, Gretton C (2007) A stochastic local search approach to vertex cover. In: Annual conference on artificial intelligence. Springer, Berlin, pp 412–426

  39. Roberto A, Maurizio B, Roberto C (2009) Optimal results and tight bounds for the maximum diversity problem. Found Comput Decis Sci 34:73–85

    Google Scholar 

  40. Rossi R, Ahmed N (2015) The network data repository with interactive graph analytics and visualization. In: AAAI, vol. 15, pp 4292–4293

  41. Rossi RA, Zhou R (2018) GraphZIP: a clique-based sparse graph compression method. J Big Data 5(1):10

    Article  Google Scholar 

  42. Shen H, Cheng X, Cai K, Hu M-B (2009) Detect overlapping and hierarchical community structure in networks. Physica A 388(8):1706–1712

    Article  Google Scholar 

  43. Shimizu S, Yamaguchi K, Masuda S (2018) A branch-and-bound based exact algorithm for the maximum edge-weight clique problem. In: International conference on computational science/intelligence and applied informatics. Springer, Cham, pp 27–47

  44. Sørensen MM (2004) New facets and a branch-and-cut algorithm for the weighted clique problem. Eur J Oper Res 154(1):57–70

    Article  MathSciNet  MATH  Google Scholar 

  45. Verma V, Aggarwal RK (2020) A comparative analysis of similarity measures akin to the Jaccard index in collaborative recommendations: empirical and theoretical perspective. Soc Netw Anal Min 10:1–16

    Article  Google Scholar 

  46. Wang X, Liu G, Li J (2017) Overlapping community detection based on structural centrality in complex networks. IEEE Access 5:25258–25269

    Article  Google Scholar 

  47. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’networks. Nature 393(6684):440

    Article  MATH  Google Scholar 

  48. Wen X, Chen WN, Lin Y, Gu T, Zhang H, Li Y, Yin Y, Zhang J (2016) A maximal clique based multiobjective evolutionary algorithm for overlapping community detection. IEEE Trans Evol Comput 21(3):363–377

    Google Scholar 

  49. Wu Q, Hao JK (2015) A review on algorithms for maximum clique problems. Eur J Oper Res 242(3):693–709

    Article  MathSciNet  MATH  Google Scholar 

  50. Žalik KR (2015) Maximal neighbor similarity reveals real communities in networks. Sci Rep 5:18374

    Article  Google Scholar 

Download references

Acknowledgements

With deep gratitude, we acknowledge the contribution of late Prof C. A. Murthy of the Indian Statistical Institute toward producing this article. He was thoroughly involved with the work contained in this article until he breathed his last. This work has been a part of the project titled Characterization and Analysis of Interaction Networks under the Women Scientist Fellowship scheme supported by the Department of Science and Technology, Government of India.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors. The research explores various theoretical aspects on weighted graph, determines maximum cliques from the sparse graph and finally detects overlapping communities from the real-life sparse network models based on our proposed modified weighted Jaccard index similarity. This paper may be an interest to the readers of your journal. We affirm that this manuscript is original, has not been published before. It is not currently being considered for publication elsewhere and has no potential conflict of interest to declare.

Author information

Authors and Affiliations

  1. Advanced Computing and Microelectronics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India

    Swati Goswami

  2. Department of Computer Science and Technology, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, 711103, India

    Swati Goswami & Asit Kumar Das

Authors
  1. Swati Goswami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Asit Kumar Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swati Goswami.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Goswami, S., Das, A.K. Determining maximum cliques for community detection in weighted sparse networks. Knowl Inf Syst 64, 289–324 (2022). https://doi.org/10.1007/s10115-021-01631-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10115-021-01631-y

Keywords

Search

Navigation