  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 the incomplete information
about the network. 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 with 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 of identifying modules in networks.
