Subcommunities and Their Mutual Relationships in a Transaction Network(Complex Networks,Proceedings of the YITP Workshop on Econophysics,Econophysics 2011-The Hitchhiker's Guide to the Economy-)

We investigate a Japanese transaction network consisting of about 800 thousand firms (nodes) and four million business relations (links) with focus on its modular structure. Communities detected by maximizing modularity often are dominated by firms with common features or behaviors in the network, such as characterized by regions or industry sectors. However, it is well known that the modularity optimization approach has a resolution limit problem, that is, it fails in identifying fine communities buried in large communities. To unfold such hidden structures, we apply the community detection to each of subnetworks formed by isolating those communities from the whole body. Subcommunities thus identified are composed of firms with finer regions, more specified sectors or business affiliations. Also we introduce a new idea of reduced modularity matrix to measure the strength of relations between (sub)communities.