Optimization Letters

, Volume 9, Issue 4, pp 615–633 | Cite as

An integer programming approach for finding the most and the least central cliques

  • Chrysafis VogiatzisEmail author
  • Alexander Veremyev
  • Eduardo L. Pasiliao
  • Panos M. Pardalos
Original Paper


We consider the problem of finding the most and the least “influential” or “influenceable” cliques in graphs based on three classical centrality measures: degree, closeness and betweenness. In addition to standard clique betweenness, which is defined as the proportion of shortest paths between any two graph nodes that pass through the clique, we also consider its optimistic and pessimistic versions, where outside nodes may favor or try to avoid shortest paths passing through the clique, respectively. We discuss the computational complexity issues for these problems and develop linear 0–1 programming formulations for each centrality metric. Finally, we demonstrate the performance of the developed formulations on real-life and synthetic networks, and provide some interesting insights based on the obtained results. In particular, our findings indicate that there are considerable variations between the centrality values of large cliques within the same networks. Moreover, the most central cliques in graphs are not necessarily the largest ones.


Clique Group centrality Integer formulation 



This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. The research was performed while the second author held a National Research Council Research Associateship Award at AFRL. The authors would like to thank the anonymous referees for their constructive comments.


  1. 1.
    COLOR02/03/04: Graph coloring and its generalizations. (2013). Accessed 9 Sept 2013
  2. 2.
    Abello, J., Pardalos, P.M., Resende, M.G.C.: On maximum clique problems in very large graphs. In: Proceedings of External Memory Algorithms, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 50, pp. 119–130. American Mathematical Society, Washington DC (1999)Google Scholar
  3. 3.
    Bassett, D.S., Owens, E.T., Daniels, K.E., Porter, M.A.: Influence of network topology on sound propagation in granular materials. Phys. Rev. E 86(4), 041306 (2012)CrossRefGoogle Scholar
  4. 4.
    Bavelas, A.: A mathematical model for group structures. Hum. Organ. 7(3), 16–30 (1948)Google Scholar
  5. 5.
    Bavelas, A.: Communication patterns in task-oriented groups. J. Acoust. Soc. Am. 22(6), 725–730 (1950)CrossRefGoogle Scholar
  6. 6.
    Boginski, V., Butenko, S., Pardalos, P.: Mining market data: a network approach. Comput. Oper. Res. 33(11), 3171–3184 (2006). doi: 10.1016/j.cor.2005.01.027 CrossRefzbMATHGoogle Scholar
  7. 7.
    Boldi, P., Vigna, S.: Axioms for centrality. arXiv preprint arXiv:1308.2140 (2013)
  8. 8.
    Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Handbook of Combinatorial Optimization, pp. 1–74. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  9. 9.
    Borgatti, S.P., Everett, M.G.: A graph-theoretic perspective on centrality. Soc. Netw. 28(4), 466–484 (2006)CrossRefGoogle Scholar
  10. 10.
    Butenko, S., Wilhelm, W.E.: Clique-detection models in computational biochemistry and genomics. Eur. J. Oper. Res. 173(1), 1–17 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Davis, T., Hu, Y.: The university of florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1–25 (2011)MathSciNetGoogle Scholar
  12. 12.
    DIMACS: 10th DIMACS implementation challenge. Available at Accessed Sept 2013 (2011)
  13. 13.
    Dolev, S., Elovici, Y., Puzis, R., Zilberman, P.: Incremental deployment of network monitors based on group betweenness centrality. Inf. Process. Lett. 109(20), 1172–1176 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ercsey-Ravasz, m, Lichtenwalter, R.N., Chawla, N.V., Toroczkai, Z.: Range-limited centrality measures in complex networks. Phys. Rev. E 85(6), 066103 (2012)CrossRefGoogle Scholar
  15. 15.
    Erdős, P., Rényi, A.: On random graphs. Publicationes Mathematicae Debrecen 6, 290–297 (1959)MathSciNetGoogle Scholar
  16. 16.
    Estrada, E.: Path Laplacian matrices: introduction and application to the analysis of consensus in networks. Linear Algebra Appl. 436(9), 3373–3391 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Everett, M.G., Borgatti, S.P.: The centrality of groups and classes. J. Math. Sociol. 23(3), 181–201 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Everett, M.G., Borgatti, S.P.: Extending centrality. Models Methods Soc. Netw. Anal. 35(1), 57–76 (2005)CrossRefGoogle Scholar
  19. 19.
    Fink, M., Spoerhase, J.: Maximum betweenness centrality: approximability and tractable cases. In: Proceedings of WALCOM: Algorithms and Computation, pp. 9–20. Springer, New York (2011)Google Scholar
  20. 20.
    Floyd, R.W.: Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962)CrossRefGoogle Scholar
  21. 21.
    Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry. 22(1), 35–41 (1977)Google Scholar
  22. 22.
    Freeman, L.C.: Centrality in social networks conceptual clarification. Soc. Netw. 1(3), 215–239 (1979)CrossRefGoogle Scholar
  23. 23.
    Gardiner, E.J., Artymiuk, P.J., Willett, P.: Clique-detection algorithms for matching three-dimensional molecular structures. J. Mol. Graph. Model. 15(4), 245–253 (1997)CrossRefGoogle Scholar
  24. 24.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman and Co., New York (1979)zbMATHGoogle Scholar
  25. 25.
    Hagberg, A.A., Schult D.A., swart, P.J.: Exploring network structure, dynamics, and function using networkx. In: Varoquaux, G., Vaught, T., Millman, J (eds) Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 1–15. Pasadena, USA (2008)Google Scholar
  26. 26.
    Hage, P., Harary, F.: Eccentricity and centrality in networks. Soc. Netw. 17(1), 57–63 (1995)CrossRefGoogle Scholar
  27. 27.
    Kolaczyk, E.D., Chua, D.B., Barthélemy, M.: Group betweenness and co-betweenness: Inter-related notions of coalition centrality. Soc. Netw. 31(3), 190–203 (2009)CrossRefGoogle Scholar
  28. 28.
    Koschützki, D., Lehmann, K.A., Peeters, L., Richter, S., Tenfelde-Podehl, D., Zlotowski, O.: Centrality indices. In: Proceedings of Network Analysis, pp. 16–61. Springer, New York (2005)Google Scholar
  29. 29.
    Krebs, V.: Uncloaking terrorist networks. First Monday 7(4) (2002). Available at Accessed 9 Sept 2013
  30. 30.
    Leavitt, H.J.: Some effects of certain communication patterns on group performance. J. Abnorm. Soc. Psychol. 46(1), 38 (1951)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Luce, R.D., Perry, A.D.: A method of matrix analysis of group structure. Psychometrika 14(2), 95–116 (1949)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Lusseau, D., Schneider, K., Boisseau, O., Haase, P., Slooten, E., Dawson, S.: The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations. Behav. Ecol. Sociobiol. 54(4), 396–405 (2003). doi: 10.1007/s00265-003-0651-y CrossRefGoogle Scholar
  33. 33.
    Pagani, G.A., Aiello, M.: The Power Grid as a complex network: A survey. Physica A: Statistical Mechanics and its Applications, 392(11), 2688–2700 (2013)Google Scholar
  34. 34.
    Puzis, R., Elovici, Y., Dolev, S.: Finding the most prominent group in complex networks. AI Commun. 20(4), 287–296 (2007)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Puzis, R., Yagil, D., Elovici, Y., Braha, D.: Collaborative attack on internet users’ anonymity. Internet Res. 19(1), 60–77 (2009)CrossRefGoogle Scholar
  36. 36.
    Reka, A., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).
  37. 37.
    Sabidussi, G.: The centrality index of a graph. Psychometrika 31(4), 581–603 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Wang, J., Peng, W., Wu, F.X.: Computational approaches to predicting essential proteins: a survey. PROTEOMICS Clin. Appl. 7(1–2), 181–192 (2013)CrossRefGoogle Scholar
  39. 39.
    Zachary, W.: An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33, 452–473 (1977)Google Scholar
  40. 40.
    Pattillo, J., Youssef, N., Butenko, S.: On clique relaxation models in network analysis. Eur. J. Oper. Res. 226(1), 9–18 (2013)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Chrysafis Vogiatzis
    • 1
    Email author
  • Alexander Veremyev
    • 2
  • Eduardo L. Pasiliao
    • 2
  • Panos M. Pardalos
    • 1
  1. 1.Department of Industrial an Systems EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Air Force Research Laboratory, Munitions DirectorateEglin AFBUSA

Personalised recommendations