Finding influential groups in networked systems: The most degree-central clique problem

Degree centrality of a cluster of vertices in a network is defined as the number of vertices outside the cluster that are adjacent to at least one vertex in the cluster. The concept of degree centrality is often used in the network analysis literature to quantify the influence of the vertex cluster within the network. That is, a large value of degree centrality shows that the cluster is adjacent to a large number of vertices outside the cluster, thus indicating its high potential for directly influencing outside components in the network. In this paper, we study the most degree-central clique problem, which is defined as the problem of finding a clique of maximum degree centrality in a network. In other words, we seek an influential cohesive cluster of vertices with no restrictions on the size of the cluster, but requiring that the cluster is highly cohesive by itself, i.e., it forms a clique. We establish that the decision version of considered problem is NP-complete. Then, we explore important theoretical properties of the problem and consequently exploit them to implement a specialized combinatorial branch-and-bound algorithm. Finally, using a collection of randomly generated and real-life networks, we compare the performance of our exact algorithm against an integer programming formulation, along with the discussion of some interesting insights.