Biased diffusion on Japanese inter-firm trading network: Estimation of   sales from network structure
  To investigate the actual phenomena of transport on a complex network, we analysed empirical data for an inter-firm trading network, which consists of about one million Japanese firms and the sales of these firms (a sale corresponds to the total in-flow into a node). First, we analysed the relationships between sales and sales of nearest neighbourhoods from which we obtain a simple linear relationship between sales and the weighted sum of sales of nearest neighbourhoods (i.e., customers). In addition, we introduce a simple money transport model that is coherent with this empirical observation. In this model, a firm (i.e., customer) distributes money to its out-edges (suppliers) proportionally to the in-degree of destinations. From intensive numerical simulations, we find that the steady flows derived from these models can approximately reproduce the distribution of sales of actual firms. The sales of individual firms deduced from the money-transport model are shown to be proportional, on an average, to the real sales. 
Hayafumi Watanabe, Hideki Takayasu, Misako Takayasu
q-fin.GN cs.SI physics.soc-ph