A Recursive Approach for Maximal (Δ, γ)-Clique Enumeration in Temporal Networks

A set of objects and a binary relation among them is often represented by a graph or network. Most of the networks that we deal with in practice (e.g., social networks, human contact networks, financial transaction networks, etc.) are time-varying in nature, i.e., the relationship changes over time. Such networks are often modeled as Temporal Networks. In this paper, we study the problem of enumerating cohesive sub-structures present in a given temporal network. We call such substructure as (Delta, gamma)-clique, which is a tuple of vertex subset and time interval, such that every pair of vertices in the vertex subset has at least gamma edges in every Delta duration of the time interval. We propose a recursive solution approach to enumerate all the maximal (Delta, gamma)-cliques. The proposed approach is divided into two parts. First, it initializes all the cliques of size 2 with maximum duration satisfying the (Delta, gamma)-clique property, and recursively, adds vertices till it becomes maximal. The correctness of the proposed method has been established, and the complexity analysis has also been done. Several experiments are carried out using real-world temporal network datasets to highlight the efficiency of the proposed approach. The reported results show that the proposed solution approach is approximately six times faster and more space-efficient than the best existing method.