To study cultural transmission is to study patterns in the way people share information, become socialized with a specific body of cultural knowledge as children, and pass on what they know. Within cultural transmission research, some folks study the underlying psychological and cognitive mechanisms, while others study the population-level consequences of those mechanisms. I study the latter, with a special focus on deep human history and longer time scales.
My premise, in this post, is that the generic structure of archaeological data places a distinctive set of requirements on transmission models meant for studying cultural transmission over long time scales. My goal is to describe why this might be, and what implications it carries for archaeologists and other historical studies of cultural transmission. In particular, I want to make the case that we need to be moving beyond well-mixed or “mean-field” mathematical models for cultural transmission when we want such models to be useful in explaining quantitative data about past cultural practices and artifact traditions.In archaeology, we typically cannot observe individual events of transmission: cases where individuals imitate one another, or when information is intentionally passed in verbal or written form. Over longer time scales, and larger spatial scales, what we can study instead is the history of the information itself. How it spreads through a population, how often new innovations occur, how long on average they persist, how much variation (say in a specific way of performing a task) is maintained in a population, and so on.
What we can observe — and this is often a different set of concepts from the theoretical quantities that we study — is the relative frequency of cultural information, measured by counting occurrences in some location, and at some point in time. There’s a subtlety here that will become quite important as I go through future posts and introduce some of my ongoing research. The act of counting is very different than measuring in some ways. When we count things, it doesn’t happen at a “point” in space or time. We always choose some chunk of both space and time in which to count.
If we are doing laboratory experiments, watching subjects imitate each other or do a “transmission chain” experiment, the block of space and time might feel small enough to think of it as a “point” in space and time, but of course it’s not really. We’re counting the relative frequency of classes of information within the testing facility, and perhaps over the course of an hour or day.
When our counting units are this tightly constrained in space and time, we often don’t need to make our mathematical models of social learning and transmission explicitly reference variation in space, we can just focus on what happens over time, perhaps by repeating our counting every day, or for several iterations of the experiment in a row. When we build a mathematical model of a social learning process — for example, prestige-biased transmission — we can develop a model based upon ordinary difference or differential equations. If we are tracking two alternative classes (or “traits”, but I’ll return in a future post to unpack this fairly fuzzy term), then a single differential equation will do (since the relative frequencies sum to 1.0, we only need to solve for the equation of motion of one of the traits, because the other equation of motion will be its mirror image). In situations where we study a set of alternatives, we typically set up a system of coupled differential equations, the solution of each being the equation of motion for a single trait out of the ensemble of alternatives.
If this sounds like the basic structure of most theoretical population genetics, that’s because it is. From Wright and Fisher to contemporary research, population geneticists most frequently study models constructed within “well-mixed” or relatively structureless populations. There are exceptions, of course. There is a rich history of spatial population genetics, beginning with Wright, and continuing through the work of Gustave Malecot, Kimura, and down to contemporary research by Slatkin, Epperson, and others.
We don’t have a comparable tradition of “spatial cultural transmission” modeling, however. And I’m arguing here that it’s high time we developed such a tradition, especially for archaeological use. Archaeologists confront a record of human history devoid of actual behavior — we confront physical objects, in a geological and spatial context. We infer (or “measure”) their position in time through the use of various dating methods, and understanding of the context of their discovery, and we directly measure fine- and coarse-grained spatial patterns in their discovery locations. We study variation in the shape, materials, design, and engineering properties of artifacts. In other words, our data consist of three axes: space, time, and form (as pointed out by Gordon Willey half a century ago).
Applying a cultural transmission model to archaeological data is fundamentally unlike applying one to observations of social learning in a living population, or in a laboratory experiment. In particular, if we do not model the population-level consequences of social learning over both time and space, then we discard much of the information we have available for distinguishing between models and hypotheses. We try to sit, effectively, on a two-legged stool. The most likely consequence is that most of our models end up giving us very similar temporal signatures, and thus can’t be discriminated in real data sets.
What does it mean to model the population consequences of transmission over both time and space?
Primarily it means dropping the mean-field approximation. Said differently, we move away from well-mixed models and explicitly model the spatial or social network distance between individuals, and analyze the population-level consequnces, and archaeological correlates, of transmission on these explicit population structures.
In practical terms, this means that transmission is modeled as being essentially local; individuals only talk to those with whom they have a social network connection, and cultural information diffuses across the network. There is a large literature attempting to model diffusion processes on social networks, which will help. Most of this literature is from economics and epidemiology, with some in statistical physics. The actual “transmission” models are fairly simple, and are mostly unbiased. So we’ll need to inject some anthropological content and modeling into the overall structures the models provide.
Analyzing diffusion models on social networks isn’t easy, even when the modes of transmission are simple. If we track the frequency of traits for each individual (say, out of a population of N) in the network, we simply end up with N coupled differential equations, and far more equations than unknowns, typically. Even if the system of ODE’s can be solved, the answers are idiosyncratic, depending upon the initial conditions. We’re after repeatable patterns here, summary statistics, not idiosyncratic solutions.
At the other extreme, we could pursue a “modified mean-field” approximation, say by classifying nodes in the network by their degree (or number of social network connections), and writing a differential equation for each degree class, and suppress other positional or graph-theoretic variation between nodes as long as they possess the same number of network edges. This approach is widely employed, following Pastor-Satorras and Vespigiani, Lopez-Pintado, and others.
We can also take a state-space correlation view, albeit a non-spatial correlation perspective, and use pair approximation to track the frequencies of trait associations among vertices, rather than trait frequencies themselves. With some effort, this approach can be extended to track the frequencies of multi-vertex “motifs” in addition to pairs.
And finally, there is a more complex approach, which treats the interacting individuals and their state as a single model — an interacting particle system — and uses the techniques of statistical physics to study the dynamic behavior of an ensemble of populations as they evolve under a transmission rule. This approach is rarely amenable, for rules and populations of any realistic complexity, to analytical solution. Instead, the statistical and spatial properties of a model’s evolutionary dynamics are derived from extensive numerical and monte-carlo simulation.
In the posts that follow, I will be examining each of these approaches, and the original mean-field approach in more detail, and charting a path forward for cultural transmission research to move beyond mean-field approximations and well-mixed models. I will be relying upon the literature in epidemic diffusion, the diffusion of innovations, an explosion of literature on statistical mechanics, and the attempts by mathematical probabilists to give rigorous results for the study of stochastic interacting systems, spin glasses and other models of disordered systems, and spatial stochastic models.
