Evolving genealogies in cultural evolution, the descendant process, and the number of cultural traits

TL;DR

This paper models the evolution of cultural traits using genealogical methods, analyzing the dynamics of ancestral lineages and trait diversity over time through a stochastic Moran-type process.

Contribution

It introduces a novel analysis of evolving cultural genealogies, linking their dynamics to the number of traits and stochastic models of ancestor and descendant counts.

Findings

Genealogical length exhibits sawtooth dynamics with random collapses.

Number of traits correlates with genealogical properties and moments.

Metastable behavior relates to the stochastic logistic model.

Abstract

We consider a Moran-type model of cultural evolution, which describes how traits emerge, are transmitted, and get lost in populations. Our analysis focuses on the underlying cultural genealogies; they were first described by Aguilar and Ghirlanda (2015) and are closely related to the ancestral selection graph of population genetics, wherefore we call them ancestral learning graphs. We investigate their dynamical behaviour, that is, we are concerned with evolving genealogies. In particular, we consider the total length of the genealogy of the entire population as a function of the (forward) time where we start looking back. This quantity shows a sawtooth-like dynamics with linear increase interrupted by collapses to near-zero at random times. We relate this to the metastable behaviour of the stochastic logistic model, which describes the evolution of the number of ancestors as well as the number of descendants of a given sample. We superpose types to the model by assuming that new inventions appear independently in every individual, and all traits of the cultural parent are transmitted to the learner in any given learning event. The set of traits of an individual then agrees with the set of innovations along its genealogy. The properties of the genealogy thus translate into the properties of the trait set of a sample. In particular, the moments of the number of traits are obtained from the moments of the total length of the genealogy.