In my previous post, I argued that cultural transmission models in archaeology [1] need to get away from being “mean-field” theories, in order to make predictions about how cultural variation is distributed in space, as well as spatiotemporally. In this post, I describe what a “mean-field” theory is, and how mean-field theories relate to a “full” description of a model’s dynamics. This post is aimed at those with a background in the basic population genetic models, or Boyd and Richerson’s cultural transmission models and their offshoots. Those with a strong background in statistical physics or spatial stochastic processes should feel free to skip it, there’s nothing here you don’t already know.
In population genetics and cultural transmission models, we often see equations which predict the evolution of a quantity over time, given parameter values and functions which describe rates of change. In other words, models for the evolution of trait frequencies tend to be difference or differential equations, depending upon whether the population is assumed to evolve continuously (overlapping generations) or discretely (as in the Wright-Fisher model). Here’s a simple example:
The above model will be most familiar to epidemiologists (and some economists), since it is a deterministic SIS epidemic model. It describes the diffusion of a single pathogen through a population, where “I” indicates the density of “infected” individuals. Not explicitly mentioned in the model is the fact that individuals begin as “S” or “susceptible,” and are infected by interaction with a previously infected individual. The final “S” indicates that individuals are infected for some period, and then become susceptible again. The greek letter beta is a parameter, indicating the transmission rate of the pathogen (per encounter); the greek letter gamma denotes the rate at which infected individuals lose the pathogen and become susceptible again.
Another common interpretation of the SIS model is unbiased social learning of a single cultural trait, where individuals also “forget” or “drop” a variant after some period of use. Alternatively, “forgetting” a trait can also simply mean that they adopt a different, unspecified cultural variant, since we are tracking only a single variant’s frequency here.
Immediately apparent in the above model is that none of the terms refer to interactions between any specific individuals. The model is written, instead, in terms of the density of infected individuals and the rate parameters. The first term is the rate at which infections increase in the population: every infective event requires a susceptible and an infected individual, so the rate of new infections is proportional to the balance between susceptibles and infectives, times the probability of transmission (beta). The second term is the rate at which infections decrease in the population: simply the density of infected individuals times the “recovery” rate (gamma). The long-term prevalence of the pathogen or variant within the population is obtained by solving for the steady-state rate (set the left-hand side to zero and solve).
A stochastic version of this mean-field model differ only in adding noise to the system due to accidents of sampling interactions between infectives and susceptibles. Whether represented fully as a Markov chain or written as a diffusion approximation, the resulting mean-field model has the same qualitative results — the process converges to a Gaussian distribution of infected individuals where the mean of the distribution is the solution to the deterministic ODE model given above, and a variance related to the population size and equilibrium number of infected individuals.
I described this model in some detail so that its structure is clear. On the face of it, there is no individual interaction here, no “microscopic dynamics” which lead to macroscopic observables. The mean-field model is written entirely in terms of the macroscopic observables. How do we get from individual interactions to this type of global, “averaged” description? How does a mean-field description arise from individual behavior?
We begin by setting up the dynamics of individual interactions. For example, in the Moran process of population genetics, transmission events proceed in continuous time. At some rate (usually a Poisson process), one individual is selected to update its state, and we want the probability of a state change to conform to the desired SIS model. We do so as follows. If the focal individual is already infected, then the probability of “recovering” and becoming susceptible again is simply the recovery rate (if we assume that recovery rates are Poisson distributed). If the focal individual is susceptible, they become infected with probability proportional to the transmission rate multiplied by the frequency of infected individuals they are in contact with. Since the Moran process proceeds at continuous time, the probability of two individuals updating in the same infinitesimal interval dt is small enough to ignore (technically, it is O(dt^2), meaning that as we shrink the time interval, the probability of > 1 event goes to zero much faster than the probability of a single event).
If, during each elemental step, each individual encounters a subset of the population (as would be the case in a real population), then the relevant probability for infection is calculated with reference to that subset. Given the stochastic nature of the Moran process, this means that each individual will face a slightly different prevalence of infectives in their subset, and thus have a different probability of infection. In other words, there will be local variation in the “neighbor infection rate.”
Since the only thing that matters (in this model) is the proportion of neighbors that are infected, we can model the neighbor infection rate as a field, to which individuals are exposed. The local strength of the field is simply the local density of infected individuals. In the full model, this field varies over time at each location within the population. The smaller the “neighbor” subset (relative to the whole population) we consider, the more spatial and temporal variation in this field. And the more difficult the challenge of characterizing the full variation in the field’s dynamics.
Conversely, the larger we make the subset with which each individual interacts, the less variable the field will be over space. As the size of the interacting subsets approaches the size of the entire population, the field simply becomes the average density of infected individuals in the population. Each individual thus interacts with a field which is the population average — hence the name “mean-field” approximation. [2]
In arriving at the mean-field model in this way, it becomes obvious how we are discarding information on variation along the way. This may be fully appropriate, especially for laboratory experiments in transmission where the experimental design has everyone interacting, or interacting with a suitably large sample of others. Or it may be appropriate for studying transmission within small social groups in living populations (although even here there will be major variation in the strength of interaction between individuals even in a small village). But when we study transmission over regional scales, we should not immediately assume that individuals equally interacted over the entire region. Everything we know about contemporary and archaeological cultures tells us otherwise.
Nearly all of the cultural transmission models in the anthropological literature today are mean-field models, including nearly all of my own previous work (apart from some spatial simulation models). This doesn’t mean that past results are wrong, merely that they discard information that can help us understand the spatial variation we see in cultural evolution.
Ultimately, my point in discussing how mean-field models arise and are structured is to understand how best to model transmission without mean-field interaction. In other words, constructing and analyzing models where individuals interact with small subsets of the total population, and thus generate spatial structure and variation. There are several methods for proceeding, and some history of doing so in epidemic and economic models of information diffusion, and a long history of spatial genetics models, so we don’t have to reinvent the wheel so much as we need to adapt these methods to our specific needs in archaeology. In upcoming posts I review some of this work and discuss more of the issues in adapting it to our needs.
NOTES:
[1] One can easily argue that spatial or non-mean-field models will be useful in other disciplines and in business contexts, but for current purposes I’m focusing on the requirements of disciplines which seek to explain a fossil record of human cultural evolution, not living populations or controlled experiments. Archaeology leads the exploration of fossil human cultural variation, so it’s a good disciplinary shorthand for historical, “deep time” empirical studies rather than contemporary cases.
[2] This description, of passing from local interaction to a limit of well-mixed, full population interaction, is also relevant to evolutionary game theory. When we examine the Nash equilibria and ESS solutions of a normal-form game, we assume random (or well-mixed) interaction within a population. The dynamics of such interactions are described by the replicator equation: a mean-field model of a game. Spatial evolutionary game theory, on the other hand, describes models where we localize interaction among smaller subsets of the population and examine the variation in equilibria and basins of attraction.
