Finding Community Structure with Performance Guarantees in Scale-Free Networks

A common property of many networks, including the Internet, biological networks, and social networks, is that the degree distribution approximately follows a power law. Such networks are found to be naturally divided into communities of densely connected nodes, known as the community structure. Despite that the existence of the community structure has been clearly indicated through experiments, there is lack of a theoretical justification for the linkage between the power-law topology and the community structure properties. Moreover, most existing community structure detection algorithms, apart from being fast and providing sub-optimal solutions, do not come with any provable solution quality. In this paper, we show, using analytical arguments, that the power-law topology indicates the presence of community structure. That is we can find in a power-law network a division of the network into communities with significant Newman´s modularity values, a well-known measure to qualify the community structure. Moreover, we provide an approximation algorithm for finding community structure via maximizing the modularity that guarantees optimal solutions up to a constant factor. To the best of our knowledge, this is the first approximation algorithm for the modularity maximization problem.