@misc{oai:niigata-u.repo.nii.ac.jp:00005277, author = {Iino, Takashi}, month = {Mar}, note = {Many kinds of complex systems that attract scientific interest can be described as networks. In general, real complex networks have nonuniform structure. They are neither regular nor random networks and in fact are composed of many clusters of nodes in networks. Such clusters called community often contain the elements that have common behavior or features in networks. Community detection for large-scale networks has been studied intensively in recent years. We investigate a Japanese transaction network consisting of about 800 thousand nodes and four million links with focus on its community structure. In the transaction network, nodes and links correspond to firms and transaction relations, respectively. The transaction network is a manifestation of economic activities. The community structure in the transaction network corresponds to groups of firms that trade actively each other. We thus expect that the community detection is a clue to understanding economical structure in Japan. To detect community structure, we employ a criterion based on modularity. The modularity evaluates the density of connections of a group for a given partition to find statistically unforeseen arrangements of edges. We search for the division which maximizes modularity to obtain an optimized community structure. Heuristic methods are usually employed because the exhaustive search of all possible divisions would be actually impossible for large-scale networks. We employ four optimization methods that are executable in a practicable time and compare the results of those community detections. The transaction network is thus separated to communities some of which are characterized by locations and industry sectors of firms. For intuitive understanding, we also visualize the transaction network using a spring-electrical model in a three-dimensional space. We regard the nodes and links as particles with identical charge and springs, respectively. It is expected that the ground state in the model, obtained by molecular dynamics simulation, provides a configuration which shows the community structure. The modularity optimization works well for identifying overall coarse-grained structure in the transaction network. There is a problem, however, known as a resolution limit of the modularity optimization. The modularity optimization fails to identify small subcommunities hidden in large communities. It is thereby possible that the communities detected in the transaction network also have nonuniform structure in a hierarchical way. To unfold such detailed structure, we analyze subcommunity structure by optimizing modularity for communities. We find some of subcommunities are characterized by regions or industry sectors in a more specific way. Furthermore, to evaluate business relationship between the subcommunities, we reduce information involved in the modularity matrix by aggregating elements in the same communities. We show that geographical distances or industrial closeness between firms play a partial role in interpreting the relationship between subcommunities. The reduced modularity matrix thus provide a helpful way to measure the strength of relations between subcommunities., 新潟大学大学院自然科学研究科, 平成23年3月23日, 新大院博(理)甲第328号, 新大院博(理)甲第328号}, title = {Study on Community and Subcommunity Structures in a Large-Scale Transaction Network}, year = {2011} }