First Stretch then Shrink and Bulk: A Two Phase Approach for Enumeration of Maximal $(\Delta, \gamma)$\mbox{-}Cliques of a Temporal Network

Abstract

A \emph{Temporal Network} (also known as \emph{Link Stream} or \emph{Time-Varying Graph}) 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 denotes the interaction between the agents $u$ and $v$ at time $t$. For analyzing the contact patterns of the agents forming a temporal network, recently the notion of classical \textit{clique} of a \textit{static graph} has been generalized as \textit{$\Delta$\mbox{-}Clique} of a Temporal Network. In the same direction, one of our previous studies introduces the notion of \textit{$(\Delta, \gamma)$\mbox{-}Clique}, which is basically a \textit{vertex set}, \textit{time interval} pair, in which every pair of the clique vertices are linked at least $\gamma$ times in every $\Delta$ duration of the time interval. In this paper, we propose a different methodology for enumerating all the maximal $(\Delta, \gamma)$\mbox{-}Cliques of a given temporal network. The proposed methodology is broadly divided into two phases. In the first phase, each temporal link is processed for constructing $(\Delta, \gamma)$\mbox{-}Clique(s) with maximum duration. In the second phase, these initial cliques are expanded by vertex addition to form the maximal cliques. From the experimentation carried out on $5$ real\mbox{-}world temporal network datasets, we observe that the proposed methodology enumerates all the maximal $(\Delta,\gamma)$\mbox{-}Cliques efficiently, particularly when the dataset is sparse. As a special case ($\gamma=1$), the proposed methodology is also able to enumerate $(\Delta,1) \equiv \Delta$\mbox{-}cliques with much less time compared to the existing methods.