Nonlinear preferential rewiring in fixed-size networks as a diffusion process

We present an evolving network model in which the total numbers of nodes and edges are conserved, but in which edges are continuously rewired according to nonlinear preferential detachment and reattachment. Assuming power-law kernels with exponents α and β , the stationary states which the degree distributions evolve toward exhibit a second-order phase transition—from relatively homogeneous to highly heterogeneous (with the emergence of starlike structures) at α=β . Temporal evolution of the distribution in this critical regime is shown to follow a nonlinear diffusion equation, arriving at either pure or mixed power laws of exponents -α and 1-α .