Emergence of network structure in models of collective evolution and evolutionary dynamics

Abstract

We consider an evolving network of a fixed number of nodes. The allocation of edges is a dynamical stochastic process inspired by biological reproduction dynamics, namely by deleting and duplicating existing nodes and their edges. The properties of the degree distribution in the stationary state is analysed by use of the Fokker–Planck equation. For a broad range of parameters, exponential degree distributions are observed. The mechanism responsible for this behaviour is illuminated by use of a simple mean field equation and reproduced by the Fokker–Planck equation. The latter is treated exactly, except for an approximate treatment of the degree–degree correlations. In the limit of 0 mutations, the degree distribution becomes a power law with exponent 1.