Exact solution for the time evolution of network rewiring models

Abstract

We consider the rewiring of a bipartite graph using a mixture of random and preferential attachment. The full mean field equations for the degree distribution and its generating function are given. The exact solution of these equations for all finite parameter values at any time is found in terms of standard functions. It is demonstrated that these solutions are an excellent fit to numerical simulations of the model. We discuss the relationship between our model and several others in the literature including examples of Urn, Backgammon, and Balls-in-Boxes models, the Watts and Strogatz rewiring problem and some models of zero range processes. Our model is also equivalent to those used in various applications including cultural transmission, family name and gene frequencies, glasses, and wealth distributions. Finally some Voter models and an example of a Minority game also show features described by our model.

Are Copying and Innovation Enough?

“…where E ′ = E. These solutions are well known in theoretical population genetics as those of the Moran model [8] and one may map the bipartite model directly onto a simple model of the genetics of a haploid population [5]. The equilibrium result for the degree distribution [3,5] is proportional to Γ (k+a)…”

“…If Π R or Π A have terms proportional to k β then this equation is exact only when β = 0 or 1 [5]. We will use the most general Π R and Π A for which (1) is exact, namely…”

“…This choice for Π A has two other special properties: one involves the scaling properties [5] and the second is that these exact equations can be solved analytically [3,5,6,4]. The generating function…”

“…One of the best ways to study the evolution of the degree distribution [5,6] is through the Homogeneity Measures, F n . This is the probability that n distinct edges chosen at random are connected to same artifact, and is given by F n (t) := (Γ (E + 1 − n)/Γ (E + 1))(d n G(z, t)/dz n ) z=1 .…”

“…From (1) we find a second order linear differential equation for each of the eigenmodes with solution [5] …”