Rich-cores in networks

Abstract: A core is said to be a group of central and densely connected nodes which governs the overall behavior of a network. Profiling this meso--scale structure currently relies on a limited number of methods which are often complex, and have scalability issues when dealing with very large networks. As a result, we are yet to fully understand its impact on network properties and dynamics. Here we introduce a simple method to profile this structure by combining the concepts of core/periphery and rich-club. The key challenge in addressing such association of the two concepts is to establish a way to define the membership of the core. The notion of a "rich-club" describes nodes which are essentially the hub of a network, as they play a dominating role in structural and functional properties. Interestingly, the definition of a rich-club naturally emphasizes high degree nodes and divides a network into two subgroups. Our approach theoretically couples the underlying principle of a rich-club with the escape time of a random walker, and a rich-core is defined by examining changes in the associated persistence probability. The method is fast and scalable to large networks. In particular, we successfully show that the evolution of the core in \emph{C. elegans} and World Trade networks correspond to key development stages and responses to historical events respectively.