Impact of hierarchical modular structure on ranking of individual nodes in directed networks

Many systems, ranging from biological and engineering systems to social systems, can be modeled as directed networks, with links representing directed interaction between two nodes. To assess the importance of a node in a directed network, various centrality measures based on different criteria have been proposed. However, calculating the centrality of a node is often difficult because of the overwhelming size of the network or because the information held about the network is incomplete. Thus, developing an approximation method for estimating centrality measures is needed. In this study, we focus on modular networks; many real-world networks are composed of modules, where connection is dense within a module and sparse across different modules. We show that ranking-type centrality measures, including the PageRank, can be efficiently estimated once the modular structure of a network is extracted. We develop an analytical method to evaluate the centrality of nodes by combining the local property (i.e. indegree and outdegree of nodes) and the global property (i.e. centrality of modules). The proposed method is corroborated by real data. Our results provide a linkage between the ranking-type centrality values of modules and those of individual nodes. They also reveal the hierarchical structure of networks in the sense of subordination (not nestedness) laid out by connectivity among modules of different relative importance. The present study raises a novel motive for identifying modules in networks.