A two-phase approach for enumeration of maximal $$(\Delta , \gamma )$$-cliques of a temporal network

A Temporal Network is often used to model a time-varying relationship among a group of agents. It is typically represented as a collection of triplets of the form (u, v, t) that denote the interaction between the agents u and v at time t. For analyzing structural patterns of such a network, the notion of (Δ,γ)-cliques has been introduced in one of our previous studies. A (Δ,γ)-clique of a temporal network is a vertex subset–time interval pair such that there exist at least γ links between every pair of vertices of the vertex set in each Δ duration of the time interval. In this paper, we propose a two-phase approach for enumerating maximal (Δ,γ)-cliques present in a temporal network. The proposed methodology is broadly divided into two phases. In the first phase, each temporal link is processed for constructing (Δ,γ)-clique(s) with maximum duration. In the second phase, these initial cliques are expanded by vertex addition to form the maximal cliques. By sequential arguments, we show that the proposed methodology correctly enumerates all the maximal (Δ,γ)-cliques. A comprehensive analysis of the running time and space requirement of the proposed methodology has been carried out. From the experimentation performed on 5 datasets, we observe that the proposed methodology enumerates all the maximal (Δ,γ)-cliques efficiently, particularly when the dataset is sparse. As a special case (γ=1), the proposed methodology is also able to enumerate (Δ,1)≡Δ-cliques in much less time compared to the existing methods.