Graph Clustering Via Intra-Cluster Density Maximization

  • Pierre MiasnikofEmail author
  • Leonidas Pitsoulis
  • Anthony J. Bonner
  • Yuri Lawryshyn
  • Panos M. Pardalos
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 315)


Graph clustering, also often referred to as network community detection, is the process of assigning common labels to vertices that are densely connected to each other but sparsely connected to the rest of the graph. There are many different approaches to clustering in the literature. However, in this article, we formulate the clustering problem as a combinatorial optimization problem. Our main contribution is a novel problem formulation that maximizes intra-cluster density, a statistically meaningful quantity. It requires the number of clusters, a softbound on cluster size and a penalty coefficient as parameter inputs. More importantly, it is designed to prevent common degeneracies, like the so-called “mega-clusters”. We end with some suggestions on numerical solution techniques and note that an ensemble-like optimization routine seems promising.


  1. 1.
    Aloise, D., Caporossi, G., Hansen, P., Liberti, L., Perron, S., Ruiz, M.: Modularity maximization in networks by variable neighborhood search. In: Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D. (eds.) Graph Partitioning and Graph Clustering, 10th DIMACS Implementation Challenge Workshop, Georgia Institute of Technology, Atlanta, GA, USA, 13–14 Feb 2012, pp. 113–128 (2012).
  2. 2.
    Bertsimas, D., Tsitsiklis, J.: Simulated annealing. Stat. Sci. 8(1), 10–15 (1993)CrossRefGoogle Scholar
  3. 3.
    Brownlee, J.: Clever Algorithms: Nature-Inspired Programming Recipes, 1st edn. (2011)Google Scholar
  4. 4.
    Creusefond, J., Largillier, T., Peyronnet, S.: Finding compact communities in large graphs. In: Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2015, ASONAM ’15, pp. 1457–1464. ACM, New York, NY, USA (2015).
  5. 5.
    van Dongen, S.: Graph clustering by flow simulation. Ph.D. thesis, Faculteit Wiskunde en Informatica, Universiteit Utrecht (2000)Google Scholar
  6. 6.
    Fan, N., Pardalos, P.M.: Linear and quadratic programming approaches for the general graph partitioning problem. J. Global Optim. 48(1), 57–71 (2010). Scholar
  7. 7.
    Fan, N., Pardalos, P.M.: Robust optimization of graph partitioning and critical node detection in analyzing networks. In: Proceedings of the 4th International Conference on Combinatorial Optimization and Applications—Volume Part I, COCOA’10, pp. 170–183. Springer, Berlin, Heidelberg (2010). Scholar
  8. 8.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75–174 (2010). Scholar
  9. 9.
    Fortunato, S., Barthélemy, M.: Resolution limit in community detection. Proc. Natl. Acad. Sci. 104(1), 36–41 (2007). Scholar
  10. 10.
    Fortunato, S., Hric, D.: Community detection in networks: a user guide. ArXiv e-prints (2016)Google Scholar
  11. 11.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, Second Edition: Data Mining, Inference, and Prediction, 2nd ed. Springer Series in Statistics. Springer (2009)Google Scholar
  12. 12.
    Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: First steps. Soc. Netw. 5(2), 109–137 (1983). Scholar
  13. 13.
    Jin, J.: Fast community detection by score. Ann. Stat. 43 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kazakovtsev, L., Antamoshkin, A.: Genetic algorithm with fast greedy heuristic for clustering and location problems. Informatica (Slovenia) 38(3) (2014).
  15. 15.
    Lancichinetti, A., Radicchi, F., Ramasco, J.J., Fortunato, S.: Finding statistically significant communities in networks. PLoS ONE 6, e18,961 (2011). Scholar
  16. 16.
    von Luxburg, U.: A Tutorial on Spectral Clustering. CoRR abs/0711.0189 (2007).
  17. 17.
    Miasnikof, P., Shestopaloff, A., Bonner, A., Lawryshyn, Y.: A statistical performance analysis of graph clustering algorithms. In: Lecture Notes in Computer Science. Springer (2018)Google Scholar
  18. 18.
    Nascimento, M., Pitsoulis, L.: Community detection by modularity maximization using GRASP with path relinking. Comput. Oper. Res. 40, 3121–3131 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E, Stat. Nonlinear, Soft Matter Phys. 69, 026,113 (2004)Google Scholar
  20. 20.
    Ovelgönne, M., Geyer-Schulz, A.: An ensemble learning strategy for graph clustering. In: Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D. (eds.) Graph Partitioning and Graph Clustering, 10th DIMACS Implementation Challenge Workshop, Georgia Institute of Technology, Atlanta, GA, USA, 13–14 Feb 2012, pp. 113–128 (2012).
  21. 21.
    Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Books on Computer Science. Dover Publications (1998).
  22. 22.
    Prokhorenkova, L.O., Prałat, P., Raigorodskii, A.: Modularity of complex networks models. In: Bonato, A., Graham, F., Prałat, P. (eds.) Algorithms and Models for the Web Graph, pp. 115–126. Springer International Publishing, Cham (2016)Google Scholar
  23. 23.
    Prokhorenkova, L.O., Prałat, P., Raigorodskii, A.: Modularity in several random graph models. In: The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB’17), Electronic Notes in Discrete Mathematics 61, 947–953 (2017). Scholar
  24. 24.
    Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach, pp. 92–115. Prentice-Hall Inc. (1995)Google Scholar
  25. 25.
    Schaeffer, S.E.: Survey: graph clustering. Comput. Sci. Rev. 1(1), 27–64 (2007). Scholar
  26. 26.
    Tasgin, M., Herdagdelen, A., Bingol, H.: Community Detection in Complex Networks Using Genetic Algorithms. ArXiv e-prints (2007)Google Scholar
  27. 27.
    Weisstein, E.: Clique. MathWorld–A Wolfram Web Resource (2018).
  28. 28.
    Yang, J., Leskovec, J.: Defining and Evaluating Network Communities Based on Ground-truth. CoRR abs/1205.6233 (2012).

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Pierre Miasnikof
    • 1
    Email author
  • Leonidas Pitsoulis
    • 2
  • Anthony J. Bonner
    • 1
  • Yuri Lawryshyn
    • 1
  • Panos M. Pardalos
    • 3
    • 4
  1. 1.University of TorontoTorontoCanada
  2. 2.Aristotle University of ThessalonikiThessalonikiGreece
  3. 3.University of FloridaGainesvilleUSA
  4. 4.National Research University HSENizhny NovgorodRussian Federation

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