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The number of cultural traits, evolving genealogies, and the descendant process

Joe Yuichiro Wakano, Hisashi Ohtsuki, Yutaka Kobayashi, Ellen Baake

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 (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 a sample of individuals from a stationary population as a function of the (forward) time at which the sample is taken. 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, or equivalently, the number of descendants of a given sample. We assume 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.

The number of cultural traits, evolving genealogies, and the descendant process

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 (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 a sample of individuals from a stationary population as a function of the (forward) time at which the sample is taken. 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, or equivalently, the number of descendants of a given sample. We assume 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.
Paper Structure (30 sections, 1 theorem, 71 equations, 15 figures)

This paper contains 30 sections, 1 theorem, 71 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The model
  3. The forward process
  4. Formal definition of the forward process
  5. The ancestral learning graph (ALG)
  6. The ALG as a stochastic process
  7. The connection with population genetics
  8. Law of large numbers
  9. Results
  10. Calculation of moments
  11. Number of cultural traits
  12. Path integrals of birth-death processes
  13. Tree length
  14. Forward simulations
  15. Procedure
  16. ...and 15 more sections

Key Result

Proposition A.1

Let $Z(0)=i \in [N]$, let $T_\ell$ be the time where the process hits state $\ell \; (1 \leq \ell \leq i)$ for the first time, and let $S_j^{(\ell)}$ be the total sojourn time in state $j \; (\ell \leq j \leq N)$ in the interval $[T_\ell, T_{\ell-1}]$. We then have where the empty product is 1.

Figures (15)

  • Figure 2: Graphical representation of a realisation of the typed model.
  • Figure 3: The ALG for the realisation of the forward model in Figure \ref{['learning-ips-innov']}, starting from $t=t_{\text{max}}$. Red and black lines are ancestral and non-ancestral, respectively, to the sample of size $n=2$ taken at $\tau=0$.
  • Figure 4: $\mathbb{E}\left[C_N\right]$ (solid) and $\mathbb{V}\left[C_N\right]$ (broken) as functions of $s$. $N=100,\mu=0.1,u=1$.
  • Figure 5: Time series $(c_{N}(t) )_{t \in [t_{\rm{max}}]_0}$ in the subcritical case (upper panel, $s=0.9$) and the supercritical case (lower panel, $s=1.3$). The last $10^4$ generations of a long run with $t_{\rm{max}}=10^5$ are shown. $N=100,\mu=0.1,u=1.$
  • Figure 6: Time series $(c_1(t) )_{t \in [t_{\rm{max}}]_0}$ (green) and $(c_N(t) )_{t \in [t_{\rm{max}}]_0}$ (blue) for the last 400 generations in a long run with $t_{\rm{max}}=10^5$ in the subcritical case (upper panel, $s=0.9$) and the supercritical case (lower panel, $s=1.3$). $N=100,\mu=0.1,u=1$.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition A.1
  • proof