  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.
