Visualizing a large-scale structure of production network by N-body simulation
📝 Abstract
Our recent study of a nation-wide production network uncovered a community structure, namely how firms are connected by supplier-customer links into tightly-knit groups with high density in intra-groups and with lower connectivity in inter-groups. Here we propose a method to visualize the community structure by a graph layout based on a physical analogy. The layout can be calculated in a practical computation-time and is possible to be accelerated by a special-purpose device of GRAPE (gravity pipeline) developed for astrophysical N-body simulation. We show that the method successfully identifies the communities in a hierarchical way by applying it to the manufacturing sector comprising tenth million nodes and a half million edges. In addition, we discuss several limitations of this method, and propose a possible way to avoid all those problems.
💡 Analysis
Our recent study of a nation-wide production network uncovered a community structure, namely how firms are connected by supplier-customer links into tightly-knit groups with high density in intra-groups and with lower connectivity in inter-groups. Here we propose a method to visualize the community structure by a graph layout based on a physical analogy. The layout can be calculated in a practical computation-time and is possible to be accelerated by a special-purpose device of GRAPE (gravity pipeline) developed for astrophysical N-body simulation. We show that the method successfully identifies the communities in a hierarchical way by applying it to the manufacturing sector comprising tenth million nodes and a half million edges. In addition, we discuss several limitations of this method, and propose a possible way to avoid all those problems.
📄 Content
Progress of Theoretical Physics Supplement 1 Visualizing a large-scale structure of production network by N-body simulation Yoshi Fujiwara NiCT/ATR CIS Applied Network Science Lab Our recent study of a nation-wide production network uncovered a community structure, namely how firms are connected by supplier-customer links into tightly-knit groups with high density in intra-groups and with lower connectivity in inter-groups. Here we propose a method to visualize the community structure by a graph layout based on a physical analogy. The layout can be calculated in a practical computation-time and is possible to be accelerated by a special-purpose device of GRAPE (gravity pipeline) developed for astrophysical N-body simulation. We show that the method successfully identifies the communities in a hierarchical way by applying it to the manufacturing sector comprising tenth million nodes and a half million edges. In addition, we discuss several limitations of this method, and propose a possible way to avoid all those problems. §1. Introduction Production network, or supplier-customer network, in economics refers to a line of economic activities in which firms buy intermediate goods from “upstream” firms, put added-value on them, and sell the goods to “downstream” firms. We recently studied a nation-wide production network comprising a million of firms and millions of supplier-customer links in Japan by applying recent statistical methods developed in complex networks (see Ref. 1)). In particular, we found that firms cluster into tightly-knit groups with high density in intra-groups and with lower connectivity in inter-groups, and that this community structure has sectoral and regional modules. In order to verify the intra-group and inter-group connectivities, we used a method of visualization of the entire manufacturing sector by a graph layout based on a physical simulation. Such a visualization of the large-scale network would be not only useful to check the community structure, but also for visualizing several influences that are taking place on the network including chain of bankruptcy, prop- agation of demand, and influence of the variation in commodity-price. In this paper, we fully explain the method of visualization, show that the re- sulting layout successfully identifies the communities. In addition, we discuss about the limitations of our method and also a possible solution. §2 briefly describes the definitions of nodes and links as firms and supplier-customer relationships, and the network to visualize. §3 compactly shows the method of community extraction and its results. Then, in §4, we explain our formulation of N-body simulation for graph drawing so as to show how the graph layout is related to the detected communities in a hierarchical way. The graph layout is an energy-based placement of nodes. This method alone has several limitations, as discussed in §5, where we shall also propose a strategy of how to avoid them. §6 summarizes the paper. arXiv:0901.2381v1 [q-fin.GN] 16 Jan 2009 2 Yoshi Fujiwara §2. A nation-wide production network Let us say that a directional link is present as A →B in a production network, where firm A is a supplier to another firm B, or equivalently, B is a customer of A. While it is difficult to record every transaction of supply and purchase among firms, it is also pointless to have a record that a firm buys a pencil from another. Necessary for our study are data of links such that the relation A →B is crucial for the activity of one or both A and B. If at least one of the firms at either end of a link rates the other firm as most important suppliers or customers, then the link should be listed. Our dataset for supplier-customer links is based on this idea. Tokyo Shoko Research, Inc., one of the leading credit research agencies in Japan, regularly gathers credit information on most of active firms. In the credit information of individual firm, suppliers and customers that are most crucial for each firm are listed up to the maximum of 24 firms respectively. We assume that the links playing important roles in the production network are recorded at either end of each link as we describe above, while we should understand that it is possible to drop relatively unimportant links from the data. We have a snapshot of production networks compiled in September 2006. In the data, the number of firms is roughly a million, and the number of directional links is more than four million. The set of nodes in the network covers essentially most of the domestic firms that are active in the sense that their credit information is required. See Ref. 1) for the study of statistical properties in the large-scale structure of the production network including scale-free degree distribution, disassortativity, correlation of degree to firm-size, low transitivity and so on, also for the relation to chains of bankruptcies taking place on the network. The global connectivity shows that basically all industries are highly entan
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