An efficient updation approach for enumerating maximal (Δ, γ)-cliques of a temporal network

Abstract

Abstract Given a temporal network $\mathcal{G}(\mathcal{V}, \mathcal{E}, \mathcal{T})$, $(\mathcal{X},[t_a,t_b])$ (where $\mathcal{X} \subseteq \mathcal{V}(\mathcal{G})$ and $[t_a,t_b] \subseteq \mathcal{T}$) is said to be a $(\Delta, \gamma)$-clique of $\mathcal{G}$, if for every pair of vertices in $\mathcal{X}$, there must exist at least $\gamma$ links in each $\Delta$ duration within the time interval $[t_a,t_b]$. Enumerating such maximal cliques is an important problem in temporal network analysis, as it reveals contact pattern among the nodes of $\mathcal{G}$. In this article, we study the maximal $(\Delta, \gamma)$-clique enumeration problem in online setting; that is, the entire link set of the network is not known in advance, and the links are coming as a batch in an iterative fashion. Suppose, the link set till time stamp $T_{1}$ (i.e. $\mathcal{E}^{T_{1}}$), and its corresponding $(\Delta, \gamma)$-clique set are known. In the next batch (till time $T_{2}$), a new set of links (denoted as $\mathcal{E}^{(T_1,T_2]}$) is arrived. Now, the goal is to update the existing $(\Delta, \gamma)$-cliques to obtain the maximal $(\Delta, \gamma)$-cliques till time stamp $T_{2}$. We formally call this problem as the Maximal $(\Delta, \gamma)$-Clique Updation Problem for enumerating maximal $(\Delta, \gamma)$-cliques. For this, we propose an efficient updation approach that can be used to enumerate maximal $(\Delta, \gamma)$-cliques of a temporal network in online setting. We show that the proposed methodology is correct, and it has been analysed for its time and space requirement. An extensive set of experiments have been carried out with four benchmark temporal network datasets. The obtained results show that the proposed methodology is efficient both in terms of time and space to enumerate maximal $(\Delta, \gamma)$-cliques in online setting. Particularly, compared to it’s off-line counterpart, the improvement caused by our proposed approach is in the order of hours and GB for computational time and space, respectively, in large dataset.