Finding disjoint dense clubs in an undirected graph

Abstract

For over a decade, software like Twitter, Facebook and WeChat have changed people's lives by creating social groups and networks easily. They give people a new convenient 'world' where we can share everything that happens around us, and social networks have grown enormously in recent years. In essence, social networks are full of data and have become an indispensable part of our life. Trust is an important feature of the relationship between two users in a social network. With the development of social networks, the trust among its members has become a big issue. In a social network, the trust among its members usually cannot be carried over many users. In the corresponding social network modeled as a graph, a user is denoted by a vertex and an edge between two vertices means that these two users communicate a lot above some threshold and they trust each other. An online social community is usually corresponding to a dense region in such a graph. A complex social network is usually composed of several groups/communities (the regions with a lot of edges), and this characterization of community structure means the appearance of densely connected groups of vertices, with only sparse connections between groups. For analyzing the structure of social networks and the relationship between users, it is important to find disjoint groups/communities with a small diameter and with a decent size, formally called dense clubs in this thesis. We focus on handling this NP-complete problem in this thesis. First, from the parameterized computational complexity point of view, we show that this problem does not admit a polynomial kernel (implying that it is unlikely to apply some reduction rules to obtain a practically small problem size). Then, we focus on the dual version of the problem, i.e., deleting 'd' vertices to obtain some disjoint dense clubs. We show that this dual problem admits a simple FPT algorithm using a bounded search tree method (the running time is still too high for practical datasets). Finally, we combine a simple reduction rule together with some heuristic methods to obtain a practical solution (verified by extensive testing on practical datasets). Empirical results show that this heuristic algorithm is very sensitive to all parameters. This algorithm is suitable on graphs which have a mixture of dense and sparse regions. These graphs are very common in the real world.