Listing All Maximal $k$-Plexes in Temporal Graphs

Many real-world networks evolve over time, that is, new contacts appear and old contacts may disappear. They can be modeled as temporal graphs where interactions between vertices (which represent people in the case of social networks) are represented by time-stamped edges. One of the most fundamental problems in (social) network analysis is community detection, and one of the most basic primitives to model a community is a clique. Addressing the problem of finding communities in temporal networks, Viard et al. [TCS 2016] introduced $\Delta$-cliques as a natural temporal version of cliques. Himmel et al. [SNAM 2017] showed how to adapt the well-known Bron-Kerbosch algorithm to enumerate $\Delta$-cliques. We continue this work and improve and extend the algorithm of Himmel et al. to enumerate temporal $k$-plexes (notably, cliques are the special case $k=1$). We define a $\Delta$-$k$-plex as a set of vertices and a time interval, where during this time interval each vertex has in each consecutive $\Delta + 1$ time steps at least one edge to all but at most $k-1$ vertices in the chosen set of vertices. We develop a recursive algorithm for enumerating all maximal $\Delta$-$k$-plexs and perform experiments on real-world social networks that demonstrate the practical feasibility of our approach. In particular, for the special case of $\Delta$-1-plexes (that is, $\Delta$-cliques), we observe that our algorithm is on average significantly faster than the previous algorithms by Himmel et al. [SNAM 2017] and Viard et al. [IPL 2018] for enumerating $\Delta$-cliques.