Enumerating maximal cliques in temporal graphs

Abstract

Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this are temporal graphs. 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] proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt to the temporal setting the Bron-Kerbosch algorithm — an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs. We obtain encouraging results both in theory (concerning worst-case 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 a significant improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien (typically two orders of magnitude).