Exact Solutions for Models of Cultural Transmission and Network Rewiring

TL;DR

This paper derives exact solutions for models of cultural transmission and network rewiring, revealing the long-term behavior and distribution patterns, including inverse power laws and homogeneity, with implications for understanding social and genetic dynamics.

Contribution

It provides the complete mean field equations and exact solutions for the evolution of degree distributions in models of cultural transmission and network rewiring.

Findings

Inverse power law distributions with a large scale cutoff are observed.

Numerical results confirm the accuracy of the analytical solutions.

Homogeneous solutions can also be reached under certain conditions.

Abstract

We look at the evolution through rewiring of the degree distribution of a network so the number edges is constant. This is exactly equivalent to the evolution of probability distributions in models of cultural transmission with drift and innovation, or models of homogeneity in genes in the presence of mutation. We show that the mean field equations in the literature are incomplete and provide the full equations. We then give an exact solution for both their long time solution and for their approach to equilibrium. Numerical results show these are excellent approximations and confirm the characteristic simple inverse power law distributions with a large scale cutoff under certain conditions. The alternative is that we reach a completely homogeneous solution. We consider how such processes may arise in practice, using a recent Minority Game study as an example.