Adapting the Bron–Kerbosch algorithm for enumerating maximal cliques in temporal graphs

Anne Sophie Himmel, Hendrik Molter, Rolf Niedermeier, Manuel Sorge

Research output: Contribution to journalArticlepeer-review

51 Scopus citations


Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this is temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating Δ-cliques, an extension of the concept of cliques to temporal graphs: for a given time period Δ, a Δ-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every Δ time steps within the time interval. Viard, Latapy, and Magnien (ASONAM 2015, TCS 2016) proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt the Bron–Kerbosch algorithm—an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs—to the temporal setting. We obtain encouraging results both in theory (concerning worst-case running time analysis based on the parameter “Δ-slice degeneracy” of the underlying graph) as well as in practice with experiments on real-world data. The latter culminates in an improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien.

Original languageEnglish
Article number35
JournalSocial Network Analysis and Mining
Issue number1
StatePublished - 1 Dec 2017


  • Community detection
  • Data science
  • Degeneracy
  • Experimental analysis
  • Fixed-parameter tractability
  • Time-varying networks
  • Δ-clique

ASJC Scopus subject areas

  • Information Systems
  • Communication
  • Media Technology
  • Human-Computer Interaction
  • Computer Science Applications


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