Updating Maximal $(\varDelta , \gamma )$-Cliques of a Temporal Network Efficiently

Suman Banerjee, Bithika Pal

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

$$(\varDelta , \gamma )$$

-clique of

$$\mathcal {G}$$

, if for every pair of vertices in

$$\mathcal {X}$$

, there must exist at least

$$\gamma $$

links in each

$$\varDelta $$

duration in

$$[t_a,t_b]$$

. In this paper, we study the maximal

$$(\varDelta , \gamma )$$

-clique enumeration problem in online setting; i.e.; the entire link set of the network is not known in advance. Suppose, the link set till time stamp

$$T_{1}$$

(i.e.,

$$\mathcal {E}^{T_{1}}$$

), and its corresponding

$$(\varDelta , \gamma )$$

-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

$$(\varDelta , \gamma )$$

-cliques to obtain the maximal

$$(\varDelta , \gamma )$$

-cliques till time stamp

$$T_{2}$$

. We formally call this problem as the Maximal

$$(\varDelta , \gamma )$$

-Clique Updation Problem. We propose an efficient updation approach and show that the proposed methodology is correct. An extensive set of experiments have been conducted with four datasets. The obtained results show that the proposed methodology is efficient both in terms of time and space to enumerate maximal

$$(\varDelta , \gamma )$$

-cliques in online setting. Compared to it’s off-line counterpart, the improvement caused by the proposed approach is in the order of hours and GB for computational time and space, respectively.