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

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.

💡 Research Summary

The paper develops a continuous‑time Moran‑type model of cultural evolution that explicitly incorporates three stochastic mechanisms: (i) death‑birth events occurring at rate u per individual, (ii) learning events at rate s per individual in which a learner randomly selects a cultural parent from the whole population, and (iii) innovation events at rate μ per individual that generate novel cultural traits. In the learning step the learner adopts all traits possessed by the chosen parent (an “all‑or‑none” transmission rule), while death erases all traits of the deceased individual.

The authors first describe the forward process using a graphical representation: horizontal lines for individuals, crosses for death‑birth, arrows for learning, and circles for innovations. This “untyped” construction captures only the timing of events. By assigning trait sets to each line and propagating them forward according to the rules above, a “typed” process is obtained, yielding the state vector (\Phi(t)={K_1(t),\dots,K_N(t),\Omega(t)}) where (K_i(t)) is the set of traits of individual i and (\Omega(t)) counts all distinct innovations up to time t.

The central novelty is the Ancestral Learning Graph (ALG), a backward‑in‑time genealogy of a sample of (n) individuals taken at a forward time (t). Starting from the sampled individuals, the algorithm traces each line backward. When a learning arrow is encountered, the graph branches, adding both the learner’s and the parent’s ancestral lines (incoming and continuing branches). When a death cross is encountered, the line is pruned. The resulting untyped ALG contains every cultural ancestor of the sample. By overlaying the forward‑time trait assignments onto the alive lines of the ALG and propagating them forward, a typed ALG is obtained.

A key observable is the total length (L(\tau)) of the ALG measured in backward time (\tau) (the sum of the durations of all branches). Because each innovation travels along a line of the genealogy, the number of distinct traits present in a sample of size n, denoted (C_n), is in one‑to‑one correspondence with the total length of the ALG: larger total length implies more opportunities for innovations to be retained. Consequently, moments of (C_n) can be derived directly from moments of (L).

The dynamics of (L(\tau)) are shown to be equivalent to a stochastic logistic process describing the joint evolution of the number of ancestors and the number of descendants. This process possesses two metastable states (high and low ancestor counts) and spends long periods near one of them before a random “collapse” to the other. In the ALG context this translates into a saw‑tooth pattern: (L(\tau)) grows approximately linearly as branches accumulate, then abruptly drops to near zero when a pruning event eliminates most lineages. The authors compute the first two moments of (L) analytically, using the known stationary distribution of the logistic process, and verify the results with extensive simulations across a range of parameters ((u,s,\mu)).

Simulation results confirm the theoretical predictions: the saw‑tooth dynamics, the distribution of collapse times, and the scaling of trait‑count moments with population size (N). The paper also explores a relaxed transmission rule where each trait is passed independently with probability (b<1); while the qualitative saw‑tooth behaviour persists, the variance of (C_n) increases, indicating that the all‑or‑none assumption amplifies the coupling between genealogy length and trait diversity.

In the discussion, the authors emphasize that the ALG provides a genealogical lens for cultural evolution, analogous to the ancestral selection graph in population genetics, but tailored to learning‑driven transmission. By linking genealogy length to cultural diversity, the work bridges stochastic demography, epidemiological logistic models, and cultural trait dynamics. Potential extensions include incorporating selective biases in learning, network‑structured interactions, and more realistic partial‑inheritance mechanisms, which would enrich the applicability of the framework to empirical cultural datasets.

Overall, the study offers a rigorous mathematical foundation for understanding how innovation, learning, and demographic turnover jointly shape the temporal fluctuations of cultural diversity, highlighting the role of metastability and genealogical collapse in producing rapid, large‑scale shifts in the cultural repertoire of populations.