Output-sensitive enumeration of maximal cliques in temporal graphs

Abstract

Community detection is one of the fundamental tasks in graph analysis, and cliques, representing a fully interconnected subset of nodes, are one of the archetypal models of community, typically focusing on those that are maximal under inclusion. Recent years witnessed an increase in prominence of temporal graphs, i.e., graphs where edges may freely appear and disappear. This scenario caused a wave of research devoted to understand the temporal structure, and how to adapt classical graph methods to this more complex model. Several adaptations have been proposed for cliques, as well as algorithms for finding maximal cliques. These, however, do not have output-sensitive guarantees, meaning considerable time could be spent to find a small number of solutions.

In this paper, we show how two different proposed models of temporal graphs, and the corresponding definitions of clique, are essentially equivalent, allowing us to consider a single model of clique without losing generality; furthermore, we develop an output-sensitive algorithm for finding all maximal cliques in the restricted scenario where each edge is present for a continuous interval: the algorithm runs in polynomial total time using polynomial space, or in incremental polynomial time if we allow exponential space usage.

While the restricted scenario limits the generality of the result, no output-sensitive algorithm for the problem exists as of yet, making this the first such result, and a first step into output-sensitive community detection in temporal graphs.