An Algorithm to Discover the k-Clique Cover in Networks

  • Luís Cavique
  • Armando B. Mendes
  • Jorge M. A. Santos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5816)


In social network analysis, a k-clique is a relaxed clique, i.e., a k-clique is a quasi-complete sub-graph. A k-clique in a graph is a sub-graph where the distance between any two vertices is no greater than k. The visualization of a small number of vertices can be easily performed in a graph. However, when the number of vertices and edges increases the visualization becomes incomprehensible. In this paper, we propose a new graph mining approach based on k-cliques. The concept of relaxed clique is extended to the whole graph, to achieve a general view, by covering the network with k-cliques. The sequence of k-clique covers is presented, combining small world concepts with community structure components. Computational results and examples are presented.


Data mining graph mining social networks 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alba, R.D.: A graph-theoretic definition of a sociometric clique. Journal of Mathematical Sociology 3, 113–126 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berners-Lee, T.: The Next Wave of the Web: Plenary Panel. In: 15th International World Wide Web Conference, WWW2006, Edinburgh, Scotland (2006)Google Scholar
  3. 3.
    Cavique, L., Luz, C.: A Heuristic for the Stability Number of a Graph based on Convex Quadratic Programming and Tabu Search, special issue of the Journal of Mathematical Sciences, Aveiro Seminar on Control Optimization and Graph Theory, Second Series (to appear, 2009)Google Scholar
  4. 4.
    Cavique, L., Rego, C., Themido, I.: A Scatter Search Algorithm for the Maximum Clique Problem. In: Ribeiro, C., Hansen, P. (eds.) Essays and Surveys in Metaheuristics, pp. 227–244. Kluwer Academic Publishers, Dordrecht (2002)CrossRefGoogle Scholar
  5. 5.
    Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cook, D.J., Holder, L.B. (eds.): Mining Graph Data. John Wiley & Sons, New Jersey (2007)zbMATHGoogle Scholar
  7. 7.
    DIMACS: Maximum clique, graph coloring, and satisfiability, Second DIMACS implementation challenge (1995), (accessed April 2009)
  8. 8.
    Erdos, P., Renyi, A.: On Random Graphs. I. Publicationes Mathematicae 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the Internet topology. In: SIGCOMM, pp. 251–262 (1999)Google Scholar
  10. 10.
    Floyd, R.W.: Algorithm 97: Shortest Path. Communications of the ACM 5(6), 345 (1962)CrossRefGoogle Scholar
  11. 11.
    Gomes, M., Cavique, L., Themido, I.: The Crew Time Tabling Problem: an extension of the Crew Scheduling Problem. Annals of Operations Research, volume Optimization in transportation 144(1), 111–132 (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Grossman, J., Ion, P., Castro, R.D.: The Erdos number Project (2007), (accessed April 2009)
  13. 13.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Science 9, 256–278 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kellerman, E.: Determination of keyword conflict. IBM Technical Disclosure Bulletin 16(2), 544–546 (1973)Google Scholar
  15. 15.
    Luce, R.D.: Connectivity and generalized cliques in sociometric group structure. Psychometrika 15, 159–190 (1950)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Milgram, S.: The Small World Problem. Psychology Today 1(1), 60–67 (1967)Google Scholar
  17. 17.
    Mokken, R.J.: Cliques, clubs and clans. Quality and Quantity 13, 161–173 (1979)CrossRefGoogle Scholar
  18. 18.
    Moreno, J.L.: Who Shall Survive? Nervous and Mental Disease Publishing Company, Washington DC (1934)Google Scholar
  19. 19.
    Scott, J.: Social Network Analysis - A Handbook. Sage Publications, London (2000)Google Scholar
  20. 20.
    Soriano, P., Gendreau, M.: Tabu search algorithms for the maximum clique. In: Johnson, D.S., Trick, M.A. (eds.) Clique, Coloring and Satisfiability, Second Implementation Challenge DIMACS, pp. 221–242 (1996)Google Scholar
  21. 21.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393(6684), 409–410 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Luís Cavique
    • 1
  • Armando B. Mendes
    • 2
  • Jorge M. A. Santos
    • 3
  1. 1.University AbertaLisboaPortugal
  2. 2.University of AzoresPonta DelgadaPortugal
  3. 3.University of EvoraEvoraPortugal

Personalised recommendations