An Efficient Updation Approach for Enumerating Maximal $(\Delta, \gamma)$\mbox{-}Cliques of a Temporal Network

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 $(\Delta, \gamma)$\mbox{-}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 paper, we study the maximal $(\Delta, \gamma)$\mbox{-}clique enumeration problem in online setting; i.e.; the entire link set of the network is not known in advance, and the links are coming as a batch in an iterative manner. 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.