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Clique Relaxations in Social Network Analysis: The Maximum k -Plex Problem

Author

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  • Balabhaskar Balasundaram

    (School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078)

  • Sergiy Butenko

    (Department of Industrial and Systems Engineering, Texas A&M University, College Station, Texas 77843)

  • Illya V. Hicks

    (Computational and Applied Mathematics Department, Rice University, Houston, Texas 77005)

Abstract

This paper introduces and studies the maximum k-plex problem , which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a k -plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.

Suggested Citation

  • Balabhaskar Balasundaram & Sergiy Butenko & Illya V. Hicks, 2011. "Clique Relaxations in Social Network Analysis: The Maximum k -Plex Problem," Operations Research, INFORMS, vol. 59(1), pages 133-142, February.
    • Handle: RePEc:inm:oropre:v:59:y:2011:i:1:p:133-142
    DOI: 10.1287/opre.1100.0851
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    1. Barabási, Albert-László & Albert, Réka & Jeong, Hawoong, 2000. "Scale-free characteristics of random networks: the topology of the world-wide web," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 281(1), pages 69-77.
    2. Réka Albert & Hawoong Jeong & Albert-László Barabási, 2000. "Error and attack tolerance of complex networks," Nature, Nature, vol. 406(6794), pages 378-382, July.
    3. Laura A. Sanchis & Arun Jagota, 1996. "Some Experimental and Theoretical Results on Test Case Generators for the Maximum Clique Problem," INFORMS Journal on Computing, INFORMS, vol. 8(2), pages 87-102, May.
    4. R. Luce & Albert Perry, 1949. "A method of matrix analysis of group structure," Psychometrika, Springer;The Psychometric Society, vol. 14(2), pages 95-116, June.
    5. S. S. Ravi & D. J. Rosenkrantz & G. K. Tayi, 1994. "Heuristic and Special Case Algorithms for Dispersion Problems," Operations Research, INFORMS, vol. 42(2), pages 299-310, April.
    6. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
    7. R. Luce, 1950. "Connectivity and generalized cliques in sociometric group structure," Psychometrika, Springer;The Psychometric Society, vol. 15(2), pages 169-190, June.
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    Cited by:

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    3. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wolfler Calvo, 2017. "A Branch-and-Price Framework for Decomposing Graphs into Relaxed Cliques," Working Papers 1723, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    4. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wolfler Calvo, 2021. "A Branch-and-Price Framework for Decomposing Graphs into Relaxed Cliques," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1070-1090, July.
    5. Foad Mahdavi Pajouh & Zhuqi Miao & Balabhaskar Balasundaram, 2014. "A branch-and-bound approach for maximum quasi-cliques," Annals of Operations Research, Springer, vol. 216(1), pages 145-161, May.
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    7. Zhou, Yi & Rossi, André & Hao, Jin-Kao, 2018. "Towards effective exact methods for the Maximum Balanced Biclique Problem in bipartite graphs," European Journal of Operational Research, Elsevier, vol. 269(3), pages 834-843.
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    14. Wayne Pullan, 2021. "Local search for the maximum k-plex problem," Journal of Heuristics, Springer, vol. 27(3), pages 303-324, June.
    15. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wolfler Calvo, 2017. "Social Network Analysis and Community Detection by Decomposing a Graph into Relaxed Cliques," Working Papers 1722, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    16. Fu, Wentao & Sun, Yang, 2021. "Rumor investigation in networks," Economic Modelling, Elsevier, vol. 98(C), pages 168-178.
    17. S. Raghavan & Rui Zhang, 2022. "Influence Maximization with Latency Requirements on Social Networks," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 710-728, March.
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