Choose a firm...
Consider the following (entirely unrealistic) scenario: suppose a firm invents a new product; they are, by definition, the only producer of the product, and if the product is new enough, there are no consumers. When there are \(N\) firms in the market making identical products, a consumer that enters at discrete time step \(t_i\) must choose between starting a new firm and producing the product herself with probability \(\rho\), and choosing to purchase the product with probability \(1-\rho\) from one of the existing firms with likelihood of choosing any particular firm proportional to the number of consumers \(x\) already purchasing the product from the firm.
The above scenario is an alternative anecdote to explain Herbert Simon’s discrete preferential attachment model, described in his 1955 Biometrika paper. Making the time steps \(t_i\) infinitesimally small and interpolating between firms turns this discrete model into a continuous one. We can write a partial differential equation to describe the density of firms \(f\) as
In a paper on which Peter Dodds and I are currently working, we analyze continuous preferential attachment processes such as this. Our findings show deep connections between preferential attachment and very general classes of probability distributions. Updates to come!