Community structure based on circular flow in a large-scale transaction network

Abstract

The objective of this study is to shed new light on the industrial flow structure embedded in microscopic supplier-buyer relations. We first construct directed networks from actual data from interfirm transaction relations in Japan; as one example, the dataset compiled by the Tokyo Shoko Research, Ltd. in 2016 contains five million links between one million firms. Then, we analyze the industrial flow structure of such a large-scale network with a special emphasis on its hierarchy and circularity. The Helmholtz-Hodge decomposition enables us to break down the flow on a directed network into two flow components: gradient flow and circular flow. The gradient flow between a pair of nodes is given by the difference of their potentials obtained by the Helmholtz-Hodge decomposition. The gradient flow runs from a node with higher potential to a node with lower potential; hence, the potential of a node shows its hierarchical position in a network. On the other hand, the circular flow component illuminates feedback loops built in a network. The potential values averaged over firms classified by the major industrial category describe hierarchical characteristics of sectors. The ordering of sectors according to the potential agrees well with the general idea of the supply chain. We also identify industrially integrated clusters of firms by applying a flow-based community detection method to the extracted circular flow network. We then find that each of the major communities is characterized by its main industry, forming a hierarchical supply chain with feedback loops by complementary industries such as transport and services.

How directed is a directed network?

“…One just has to bear in mind that often we consider our weights w to be flows, so the terminology can be confusing. They are the same in the main case treated in our paper (all targets = 1) and in [ 35 ] ( w = F ), but in general need distinguishing. The approach of Kichikawa et al .…”

“…[ 34 ] and Kichikawa et al . [ 35 ] obtain levels and a notion of circularity by a graph version of the Helmholtz–Hodge decomposition of a vector field into a gradient part and a conservative (divergence-free) part. Here we show the equivalence of our method to theirs.…”

“…The approach of Kichikawa et al . [ 35 ] leads to alternative terminology, which can be advantageous. Their ‘circularity’ is our incoherence.…”

Graph Hierarchy: A novel framework to analyse hierarchical structures in complex networks

“…There are some approaches with a similar stratagem that have attempted to quantify hierarchy by generalising trophic analysis, see 33 and 34 . These approaches however cannot differentiate vertices in the same strongly connected component and they do not have any explicit links to the topology of the graph.…”

The sustainability and the survivability of Kyoto’s traditional craft industry revealed from supplier-customer network

“…• OTHER: nodes not connected to G � Each component makes different contributions to the network in terms of economic activity, while the SCC plays a central role here. All industries in the specific economy are closely connected, and the growth of the core industries may have greater significance for the economy's growth compared with that of the peripheral industries [29].…”