By analysing the financial data of firms across Japan, a nonlinear power lawwith an exponent of 1.3 was observed between the number of business partners(i.e. the degree of the inter-firm trading network) and sales. In a previousstudy using numerical simulations, we found that this scaling can be explainedby both the money-transport model, where a firm (i.e. customer) distributesmoney to its out-edges (suppliers) in proportion to the in-degree ofdestinations, and by the correlations among the Japanese inter-firm tradingnetwork. However, in this previous study, we could not specifically identifywhat types of structure properties (or correlations) of the network determinethe 1.3 exponent. In the present study, we more clearly elucidate therelationship between this nonlinear scaling and the network structure byapplying mean-field approximation of the diffusion in a complex network to thismoney-transport model. Using theoretical analysis, we obtained the mean-fieldsolution of the model and found that, in the case of the Japanese firms, thescaling exponent of 1.3 can be determined from the power law of the averagedegree of the nearest neighbours of the network with an exponent of -0.7.
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