Econophysics on networks

Krugman [“The Self-Organizing Economy“, Blackwell, 1996] proposed a continuous model, in time and space, for the emergence of polycentric urban areas in the regional space. By using a discrete version of the Krugman model on spatial networks, we predicted the distribution of jobs among the different localities inside several economic regions in Ohio and Texas [S. Kaufman, M. Kaufman, M. Salling, Applied Network Science, 2019]. The model predicts new spatial distribution of jobs that emerges from the current fractions of jobs at time t and location x: n_t,x  through interactions among localities in a region. The market potential function of any location x at time t is P_t,x = Σ_y q_x,yn_t,y .  It is determined by a network matrix q_x,y connecting any two locations in the region.  Employment gradually moves towards locations considered relatively attractive if their market potential is above the spatial average:       P_t,x > <P_t >.  Similarly, jobs move away from locations with below-average market potential.  I will discuss a couple of general properties of the model.  First, I will determine the stationary distribution and its stability. Second, I will show that the market potential, which is analogous to the fitting function, satisfies the Fisher [“The genetical theory of natural selection”, Oxford, 1930] equation of natural selection:  <P_t+1> - <P_t> = var(P_t,x).