Diffusion Models of Cultural Evolution in Social Networks

This paper describes an ongoing effort to model cultural evolution on a social network with fractional diffusion equations, parallel to how biological evolution is modeled with ordinary diffusion equations. Such modeling with diffusion equations would make it possible to calculate probabilities of spread of cultural innovations as well as to calculate how quickly such spread can happen. Because of the heavy-tailed connectivity of social networks, the second derivative found in the traditional diffusion equations needs to be replaced with an appropriate fractional derivative. It is well-understood how fractional diffusion equations can be derived for ordinary "anomalous" diffusion problems, but how to do this for cultural evolution is still an unsolved problem, because for cultural evolution the rate of diffusion depends on the frequency of a cultural item in the population. Here preliminary results and simulation results are presented, and qualitative differences between the biological and cultural case are pointed out.