Enumerating Isolated Cliques in Temporal Networks

  • Hendrik MolterEmail author
  • Rolf Niedermeier
  • Malte Renken
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 882)


Isolation is a concept from the world of clique enumeration that is mostly used to model communities that do not have much contact to the outside world. Herein, a clique is considered isolated if it has few edges connecting it to the rest Motivated by recent work on enumerating cliques in temporal networks, we bring the isolation concept to this setting. We discover that the addition of the time dimension leads to six distinct natural isolation concepts. Our main contribution is the development of fixed-parameter enumeration algorithms for five of these six clique types employing the parameter “degree of isolation”. On the empirical side, we implement and test these algorithms on (temporal) social network data, obtaining encouraging preliminary results.


Community detection Dense subgraphs Social network analysis Time-evolving data Fixed-parameter tractability 



We want to thank our student assistant Fabian Jacobs for his work on the implementation of our algorithms and anonymous reviewers for helpful feedback.


  1. 1.
    Bentert, M., Himmel, A.S., Molter, H., Morik, M., Niedermeier, R., Saitenmacher, R.: Listing all maximal \(k\)-plexes in temporal graphs. ACM J. Exp. Algorithm. 24(1), 1.13:1–1.13:27 (2019)MathSciNetGoogle Scholar
  2. 2.
    van Bevern, R., Fluschnik, T., Mertzios, G.B., Molter, H., Sorge, M., Suchý, O.: The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs. Discret. Optim. 30, 20–50 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chechik, S., Johnson, M.P., Parter, M., Peleg, D.: Secluded connectivity problems. Algorithmica 79(3), 708–741 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eppstein, D., Strash, D.: Listing all maximal cliques in large sparse real-world graphs. ACM J. Exp. Algorithm. 18, 1–3 (2013)zbMATHGoogle Scholar
  5. 5.
    Fournet, J., Barrat, A.: Contact patterns among high school students. PLoS ONE 9(9), 1–17 (2014)CrossRefGoogle Scholar
  6. 6.
    Génois, M., Barrat, A.: Can co-location be used as a proxy for face-to-face contacts? EPJ Data Sci. 7(1), 11 (2018)CrossRefGoogle Scholar
  7. 7.
    Himmel, A.S., Molter, H., Niedermeier, R., Sorge, M.: Adapting the Bron-Kerbosch algorithm for enumerating maximal cliques in temporal graphs. Soc. Netw. Anal. Min. 7(1), 35:1–35:16 (2017)CrossRefGoogle Scholar
  8. 8.
    Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519(3), 97–125 (2012)CrossRefGoogle Scholar
  9. 9.
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Isolation concepts for clique enumeration: comparison and computational experiments. Theor. Comput. Sci. 410(52), 5384–5397 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ito, H., Iwama, K.: Enumeration of isolated cliques and pseudo-cliques. ACM Trans. Algorithms 5(4), 40:1–40:21 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Komusiewicz, C., Hüffner, F., Moser, H., Niedermeier, R.: Isolation concepts for efficiently enumerating dense subgraphs. Theor. Comput. Sci. 410(38–40), 3640–3654 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Latapy, M., Viard, T., Magnien, C.: Stream graphs and link streams for the modeling of interactions over time. Soc. Netw. Anal. Min. 8(1), 61 (2018)CrossRefGoogle Scholar
  13. 13.
    Michail, O.: An introduction to temporal graphs: an algorithmic perspective. Internet Math. 12(4), 239–280 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rossetti, G., Cazabet, R.: Community discovery in dynamic networks: a survey. ACM Comput. Surv. 51(2), 35 (2018)CrossRefGoogle Scholar
  15. 15.
    Viard, T., Latapy, M., Magnien, C.: Computing maximal cliques in link streams. Theor. Comput. Sci. 609, 245–252 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Viard, T., Magnien, C., Latapy, M.: Enumerating maximal cliques in link streams with durations. Inf. Process. Lett. 133, 44–48 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Hendrik Molter
    • 1
    Email author
  • Rolf Niedermeier
    • 1
  • Malte Renken
    • 1
  1. 1.Faculty IV, Algorithmics and Computational ComplexityTU BerlinBerlinGermany

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