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Distance-Based Clique Relaxations in Networks: s-Clique and s-Club

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Models, Algorithms, and Technologies for Network Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 59))

Abstract

The concept of the clique, originally introduced as a model of a cohesive subgroup in the context of social network analysis, is a classical model of a cluster in networks. However, the ideal cohesiveness properties guaranteed by the clique definition put limitations on its applicability to situations where enforcing such properties is unnecessary or even prohibitive. Motivated by practical applications of diverse origins, numerous clique relaxation models, which are obtained by relaxing certain properties of a clique, have been introduced and studied by researchers representing different fields. Distance-based clique relaxations, which replace the requirement on pairwise distances to be equal to 1 in a clique with less restrictive distance bounds, are among the most important such models. This chapter surveys the up-to-date progress made in studying two common distance-based clique relaxation models called s-clique and s-club, as well as the corresponding optimization problems.

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References

  1. Abello, J., Resende, M.G.C., Sudarsky, S.: Massive quasi-clique detection. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics, pp. 598–612. Springer, London (2002)

    Chapter  Google Scholar 

  2. Adler, N.: Competition in a deregulated air transportation market. Eur. J. Oper. Res. 129, 337–345 (2001)

    Article  MATH  Google Scholar 

  3. Alba, R.D.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3, 113–126 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Almeida, M.T., Carvalho, F.D.: Integer models and upper bounds for the 3-club problem. Networks 60, 155–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arora, S., Lund, C., Motwani, R., Szegedy, M.: Proof verification and hardness of approximation problems. J. ACM 45, 501–555 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45(1), 70–122 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Asahiro, Y., Miyano, E., Samizo, K.: Approximating maximum diameter-bounded subgraphs. In: Proceedings of the 9th Latin American Conference on Theoretical Informatics (LATIN’10), pp. 615–626. Springer, Berlin (2010)

    Chapter  Google Scholar 

  8. Bader, J.S., Chaudhuri, A., Rothberg, J.M., Chant, J.: Gaining confidence in high-throughput protein interaction networks. Nat. Biotechnol. 22, 78–85 (2004)

    Article  Google Scholar 

  9. Balasundaram, B., Butenko, S.: Graph domination, coloring and cliques in telecommunications. In: Resende, M.G.C., Pardalos, P.M. (eds.) Handbook of Optimization in Telecommunications, pp. 865–890. Springer, New York (2006)

    Chapter  Google Scholar 

  10. Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10, 23–39 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Batagelj, V.: Networks/Pajek graph files. http://vlado.fmf.uni-lj.si/pub/networks/pajek/data/gphs.htm (2005). Accessed July 2013

  12. Berry, N., Ko, T., Moy, T., Smrcka, J., Turnley, J., Wu, B.: Emergent clique formation in terrorist recruitment. In: The AAAI-04 Workshop on Agent Organizations: Theory and Practice, July 25, 2004, San Jose, California (2004). http://www.cs.uu.nl/~virginia/aotp/papers.htm

    Google Scholar 

  13. Bollobás, B.: Extremal Graph Theory. Academic Press, New York (1978)

    MATH  Google Scholar 

  14. Bollobás, B.: Random Graphs. Academic Press, New York (1985)

    MATH  Google Scholar 

  15. Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 1–74. Kluwer Academic, Dordrecht (1999)

    Chapter  Google Scholar 

  16. Bourjolly, J.-M., Laporte, G., Pesant, G.: Heuristics for finding k-clubs in an undirected graph. Comput. Oper. Res. 27, 559–569 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bourjolly, J.-M., Laporte, G., Pesant, G.: An exact algorithm for the maximum k-club problem in an undirected graph. Eur. J. Oper. Res. 138, 21–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brélaz, D.: New methods to color the vertices of a graph. Commun. ACM 22(4), 251–256 (1979)

    Article  MATH  Google Scholar 

  19. Buchanan, A., Sung, J.S., Butenko, S., Boginski, V., Pasiliao, E.: On connected dominating sets of restricted diameter. Working paper (2012)

    Google Scholar 

  20. Butenko, S., Wilhelm, W.: Clique-detection models in computational biochemistry and genomics. Eur. J. Oper. Res. 173, 1–17 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Carraghan, R., Pardalos, P.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382 (1990)

    Article  MATH  Google Scholar 

  22. Carvalho, F.D., Almeida, M.T.: Upper bounds and heuristics for the 2-club problem. Eur. J. Oper. Res. 210(3), 489–494 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chang, M.S., Hung, L.J., Lin, C.R., Su, P.C.: Finding large k-clubs in undirected graphs. In: Proc. 28th Workshop on Combinatorial Mathematics and Computation Theory (2011)

    Google Scholar 

  24. Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci. 72, 1346–1367 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86, 165–177 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dekker, A., Pérez-Rosés, H., Pineda-Villavicencio, G., Watters, P.: The maximum degree & diameter-bounded subgraph and its applications. J. Math. Model. Algorithms 11, 249–268 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dekker, A.H.: Social network analysis in military headquarters using CAVALIER. In: Proceedings of Fifth International Command and Control Research and Technology Symposium, Canberra, Australia, pp. 24–26 (2000)

    Google Scholar 

  28. Diestel, R.: Graph Theory. Springer, Berlin (1997)

    MATH  Google Scholar 

  29. Dimacs. Second Dimacs Implementation Challenge. http://dimacs.rutgers.edu/Challenges/ (1995). Accessed July 2013

  30. Dimacs. Tenth Dimacs Implementation Challenge. http://cc.gatech.edu/dimacs10/. Accessed July 2013

  31. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  32. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  33. Gendreau, M., Soriano, P., Salvail, L.: Solving the maximum clique problem using a tabu search approach. Ann. Oper. Res. 41, 385–403 (1993)

    Article  MATH  Google Scholar 

  34. Goffman, C.: And what is your Erdös number? Am. Math. Mon. 76, 791 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  35. Grossman, J., Ion, P., De Castro, R.: The Erdös Number Project. http://www.oakland.edu/enp/ (1995). Accessed July 2013

  36. Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. In: Thilikos, D., Woeginger, G. (eds.) Parameterized and Exact Computation. Lecture Notes in Computer Science, vol. 7535, pp. 231–241. Springer, Berlin (2012)

    Chapter  Google Scholar 

  37. Hartung, S., Komusiewicz, C., Nichterlein, A.: On structural parameterizations for the 2-club problem. In: Proceedings of the 39th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM ’13). Lecture Notes in Computer Science, vol. 7741, pp. 233–243. Springer, Berlin (2013)

    Google Scholar 

  38. Jaillet, P., Song, G., Yu, G.: Airline network design and hub location problems. Location Sci. 4, 195–212 (1996)

    Article  MATH  Google Scholar 

  39. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Chapter  Google Scholar 

  40. Kim, D., Wu, Y., Li, Y., Zou, F., Du, D.-Z.: Constructing minimum connected dominating sets with bounded diameters in wireless networks. IEEE Trans. Parallel Distrib. Syst. 20(2), 147–157 (2009)

    Article  Google Scholar 

  41. Luce, R.D.: Connectivity and generalized cliques in sociometric group structure. Psychometrika 15, 169–190 (1950)

    Article  MathSciNet  Google Scholar 

  42. Luce, R.D., Perry, A.D.: A method of matrix analysis of group structure. Psychometrika 14, 95–116 (1949)

    Article  MathSciNet  Google Scholar 

  43. Mahdavi Pajouh, F.: Polyhedral combinatorics, complexity & algorithms for k-clubs in graphs. PhD thesis, Oklahoma State University (July 2012)

    Google Scholar 

  44. Mahdavi Pajouh, F., Balasundaram, B.: On inclusionwise maximal and maximum cardinality k-clubs in graphs. Discrete Optim. 9(2), 84–97 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Marinček, J., Mohar, B.: On approximating the maximum diameter ratio of graphs. Discrete Math. 244, 323–330 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Memon, N., Kristoffersen, K.C., Hicks, D.L., Larsen, H.L.: Detecting critical regions in covert networks: a case study of 9/11 terrorists network. In: The Second International Conference on Availability, Reliability and Security, pp. 861–870 (2007)

    Chapter  Google Scholar 

  47. Miao, J., Berleant, D.: From paragraph networks to document networks. In: Proceedings of the International Conference on Information Technology: Coding and Computing (ITCC 2004), vol. 1, pp. 295–302 (2004)

    Google Scholar 

  48. Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mokken, R.J.: Cliques, clubs and clans. Qual. Quant. 13, 161–173 (1979)

    Article  Google Scholar 

  50. Newman, M.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  51. Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120, 197–207 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  52. Pasupuleti, S.: Detection of protein complexes in protein interaction networks using n-clubs. In: Proceedings of the 6th European Conference on Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics. Lecture Notes in Computer Science, vol. 4973, pp. 153–164. Springer, Berlin (2008)

    Chapter  Google Scholar 

  53. Pattillo, J., Wang, Y., Butenko, S.: On distance-based clique relaxations in unit disk graphs. Working paper (2012)

    Google Scholar 

  54. Pattillo, J., Youssef, N., Butenko, S.: On clique relaxation models in network analysis. Eur. J. Oper. Res. (2012). doi:10.1016/j.ejor.2012.10.021

    Google Scholar 

  55. Rothenberg, R.B., Potterat, J.J., Woodhouse, D.E.: Personal risk taking and the spread of disease: beyond core groups. J. Infect. Dis. 174(Supp. 2), S144–S149 (1996)

    Article  Google Scholar 

  56. Sageman, M.: Understanding Terrorist Networks. University of Pennsylvania Press, Philadelphia (2004)

    Google Scholar 

  57. Sampson, R.J., Groves, B.W.: Community structure and crime: testing social-disorganization theory. Am. J. Sociol. 94, 774–802 (1989)

    Article  Google Scholar 

  58. Schäfer, A.: Exact algorithms for s-club finding and related problems. Master’s thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena (2009)

    Google Scholar 

  59. Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6, 883–891 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  60. Scott, J.: Social Network Analysis: A Handbook, 2nd edn. Sage, London (2000)

    Google Scholar 

  61. Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5, 269–287 (1983)

    Article  MathSciNet  Google Scholar 

  62. Seidman, S.B., Foster, B.L.: A graph theoretic generalization of the clique concept. J. Math. Sociol. 6, 139–154 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  63. Shahinpour, S., Butenko, S.: Algorithms for the maximum k-club problem in graphs. J. Comb. Optim. (2013). doi:10.1007/s10878-012-9473-z

    MathSciNet  MATH  Google Scholar 

  64. Terveen, L., Hill, W., Amento, B.: Constructing, organizing, and visualizing collections of topically related web resources. ACM Trans. Comput.-Hum. Interact. 6, 67–94 (1999)

    Article  Google Scholar 

  65. Veremyev, A., Boginski, V.: Identifying large robust network clusters via new compact formulations of maximum k-club problems. Eur. J. Oper. Res. 218(2), 316–326 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  66. Wasserman, S., Faust, K.: Social Network Analysis. Cambridge University Press, New York (1994)

    Google Scholar 

  67. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3, 103–128 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Partial support of Air Force Office of Scientific Research (Award FA9550-12-1-0103) and the U.S. Department of Energy (Award DE-SC0002051) is gratefully acknowledged. We thank Baski Balasundaram and Foad Mahdavi Pajouh for their comments on a preliminary version of the paper, which helped to improve the presentation.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843-3131, USA

    Shahram Shahinpour & Sergiy Butenko

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  1. Shahram Shahinpour
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  2. Sergiy Butenko
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Correspondence to Sergiy Butenko .

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Editors and Affiliations

  1. Dept. of Industrial and Systems Engineer, University of Florida, Gainesville, Florida, USA

    Boris I. Goldengorin

  2. Department of Applied Mathematics, National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Valery A. Kalyagin

  3. Department of Industrial & Systems Engin, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Shahinpour, S., Butenko, S. (2013). Distance-Based Clique Relaxations in Networks: s-Clique and s-Club. In: Goldengorin, B., Kalyagin, V., Pardalos, P. (eds) Models, Algorithms, and Technologies for Network Analysis. Springer Proceedings in Mathematics & Statistics, vol 59. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8588-9_10

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