Adapting the Bron-Kerbosch Algorithm for Enumerating Maximal Cliques in   Temporal Graphs
  Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this are 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 delta-cliques, an extension of the concept of cliques to temporal graphs: for a given time period delta, a delta-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 delta time steps within the time interval. Viard, Latapy, and Magnien [ASONAM 2015, TCS 2016] proposed a greedy algorithm for enumerating all maximal delta-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 "delta-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 delte-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien. 
Anne-Sophie Himmel and Hendrik Molter and Rolf Niedermeier and Manuel   Sorge
cs.DS cs.CC cs.SI