Bipartite producer–consumer networks and the size distribution of firms

A bipartite producer–consumer network is constructed to describe the industrial structure. The edges from consumer to producer represent the choices of the consumer for the final products and the degree of producer can represent its market share. So the size distribution of firms can be characterized by producer's degree distribution. The probability for a producer receiving a new consumption is determined by its competency described by initial attractiveness and the self-reinforcing mechanism in the competition described by preferential attachment. The cases with constant total consumption and with growing market are studied. The following results are obtained: (1) Without market growth and a uniform initial attractiveness a  , the final distribution of firm sizes is Gamma distribution for a>1a>1 and is exponential for a=1a=1. If a<1a<1, the distribution is power in small size and exponential in upper tail. (2) For a growing market, the size distribution of firms obeys the power-law. The exponent is affected by the market growth and the initial attractiveness of the firms.