## Renormalization Theory and Cultural Transmission in Archaeology

Hopefully a long hiatus is behind me, and I’ll be posting more regularly about research topics and scientific issues.  I spent much of the last academic year on conference papers, a dissertation proposal, and getting myself positioned for general exams this year.  With that accomplished, and my dissertation research solidified and underway, I feel better able to post on my research in more detail.

In general, my topic concerns the “renormalization” of cultural transmission models.  This terminology will probably be unfamiliar to anthropologists and social scientists, so I’m not going to emphasize the term or formal renormalization theory in upcoming publications or my dissertation, but it is absolutely what I’m studying.  I thought a blog post would be a good place to describe this concept, and its relationship to concepts more familiar to anthropologists.

Those who study long-term records of behavior or evolution face the problem that evolutionary models are individual-based or “microevolutionary,” and describe the detailed change in adoption of traits or flow of genetic information within a population, while our empirical data describe highly aggregated, temporally averaged counts or frequencies.  This mismatch in temporal scales is extreme enough that the “evolutionary synthesis” of the 1940′s tended to separate consideration of “microevolution” and “macroevolution” into different sets of processes (largely as a result of George Gaylord Simpson’s pioneering work).  The study of the fossil record of life on earth has rightly focused mainly on the phylogenetic history of species and higher taxa, in their paleoecological contexts.  When studying the archaeological record of human behavior over shorter (albeit still substantial) time scales, it seems less clear that microevolutionary models cannot inform our explanations.

At the same time, our usage of microevolutionary models of cultural transmission, to date, has almost universally ignored the vast difference in time scales between our evidence and the behavioral events and “system states” we model.  The sole exception to this rule, actually, seems to be Fraser Neiman’s 1990 dissertation, which has a sophisticated discussion of the effects of time-averaging on neutral models and cultural trait frequencies.  So,  an important question would be:  what do cultural transmission models look like, when we view their behavior through the lens of a much longer-term data source?

This is precisely the kind of question that renormalization theory answers, as formulated in physics.  Below the jump, I describe renormalization in more detail.

## Open Problem: Can we detect modes of transmission within heterogeneous populations?

Since Bentley and Shennan’s work demonstrating that random copying processes generate power-law frequency spectra, a significant thread in cultural transmission research has focused on the shape of frequency distributions.  In my previous post, I cited Mesoudi and Lycett’s (2009) paper in passing, and in this post I want to highlight an issue that constitutes an important open problem in transmission modeling.

Mesoudi and Lycett note (p. 42) in passing that “perhaps some mix of conformity, anti-conformity, and innovation combine to produce aggregate, population-level data that are indistinguishable from random copying.”  The authors go on to note that this claim has not been tested explicitly, and I believe as of this writing (Dec 2010), that this still constitutes an open issue.

## CT: “Random Copying” is not just “Cultural Drift”

I’ve been re-reading a lot of the cultural transmission literature lately, in preparation for a writing project, and anthropologists (including archaeologists) working on CT tend to discuss unbiased transmission (or random copying, to use Bentley’s term) and drift as if they referred to the same thing.

They don’t.

For example, in their superb article “Random copying, frequency-dependent copying and culture change,” Alex Mesoudi and Stephen Lycett say:  ”In recent years, several studies have … proposed that the frequency distributions of various cultural traits … can be explained using a simple model of random copying, the cultural analogue of genetic drift.” (p. 41-42, references omitted for clarity, italics in original).   I use Mesoudi and Lycett’s quote because it is particularly clear in drawing this parallel, but one can find similar statements throughout many other works on cultural transmission, particularly since Bentley’s work on power-law frequency distributions.

The problem is, “random copying” and “drift” have nothing to do with one another, except possibly the statistical properties of their effects upon a well-mixed population.

## Generalized evolutionary models incorporating state, environment, and structure

Over the last few months, a high-profile controversy has been brewing in evolutionary biology. Martin Nowak, Corina Tarnita, and E.O. Wilson published “The Evolution of Eusociality” in Nature, in which they apply Nowak and Tarnita’s work on evolutionary set theory to the evolution of cooperation and particularly eusociality among the social insects. What made this work controversial is their claim that such an approach renders inclusive fitness theory unncessary. But what got legions of evolutionary biologists (including Alan Grafen) really hot under the collar was the additional suggestion that inclusive fitness makes enough simplifying assumptions that it doesn’t even apply to the empirical cases which it is purported to best explain, potentially calling into question a great deal of work based on IF theory.

I’m not qualified to evaluate the latter claims, which is fine because Alan Grafen and Richard Dawkins are on the warpath and I’m sure we’ll see a paper in response quite soon.

I’m more interested in the general claim, that the approach taken by Nowak et al. represents a useful and general way of looking at evolution in realistically structured populations. Because I think they’re on the right track. The last thirty years have seen an explosion of evolutionary models for populations structured in various ways, because virtually everyone now realizes the stability of cooperative phenomena depend crucially upon assortative interaction.  In other words, structured interaction helps keep defectors from invading groups of mutually supporting cooperators.  Some such groups are kin-based, others are based upon social network connections, and still other groupings are spatial.  All of these situations can be described by understanding evolutionary dynamics upon generalized networks or graphs (since spatial lattices are simply regular graphs).

And understanding the effect of complex and rich structure upon evolutionary dynamics is critical, as a growing mountain of theoretical work has shown. We started understanding evolution in quantitative, dynamical system terms (with the work of Wright and Fisher), by largely ignoring interaction structure (although Wright did some crucial early work on assortative mating). Theoretical biologists employed what physicists call a “mean-field approximation,” assuming that every organism if a population is equally likely to reproduce with any other, and thus evolutionary forces can be treated as an average “field” applied to the state of the population as a whole.1 Nearly every equation you see in a basic text on population genetics is a mean-field model. The same is true for quantitative models of social learning 2 Boyd and Richerson’s (1985) landmark book is filled with mean-field models, and quite understandably so.  Mean-field models are where we typically start trying to understand a complex phenomenon.

Over the last decade or more, Martin Nowak and his group have been key contributors to understanding how the dynamics of evolutionary processes depend upon relaxing the mean-field approximation and incorporating explicitly the structure of interaction into our models.  But even what we now call “complex” network models tend to represent only a single type of relationship between individuals. The “complex” moniker here refers to topology, not richness of association or relationship. So I find Nowak and Tarnita’s work on “evolutionary set theory” quite interesting, as a generalization of the network concept (and which clearly can interoperate with it).  In this posting, I want to explore where such an approach leads, in terms of the structure of evolutionary models, and what methods will be required to analyze those models as we add realism and complexity.

## Why I buy the “niche construction” argument for evolutionary biology

Carl Lipo and I were recently talking about a recent paper by Kevin Laland and Michael J. O’Brien, Niche Construction Theory and Archaeology, and it stimulated me to think about why I buy the argument that niche construction theory (NCT) is important for the future development of evolutionary theory.

After all, scientific theory has “fads” like anything else, one could argue (in parallel to the argument recently developed by Nowak and colleagues concerning inclusive fitness theory) that any “niche construction” argument can also be formulated in a different framework by simply writing standard natural selection models, with appropriate values or operators for fitness values.

I believe that while Nowak et al. are absolutely on the right track with respect to population structure, cooperation, and eusociality, that NCT arguments cannot always be reduced to an equivalent “traditional selection model.” To see why, we need to follow Richard Lewontin’s argument from a 1982 and 1983 paper originally defining NCT.

Lewontin, as he so often has in evolutionary biology, stripped the argument down to its essentials and provided a very simple skeleton. In this case, he boils down evolutionary biology to an ansatz or generic model as follows:

\begin{eqnarray}
\frac{dO}{dt} &=& f(O, E) \\
\frac{dE}{dt} &=& g(E) \\
\end{eqnarray}

The first equation describes the evolution of a population by natural selection as a dynamical system, in which rate of change of the population state (O), is given by a function f whose values depend both upon the population state itself and the nature of the environment (E). This dynamical system is fully generic and can describe constant selection (when the function f ignores O and only depends upon E, for example), or frequency-dependent selection (when the function f depends mostly upon the population state, with the environment providing “background fitness” to the payoffs of a particular evolutionary game. And so on….density dependence fits in this model as well.

Simple or toy models of evolutionary processes might focus only on the first equation. But we also know, in the real world, that the environment itself is changing. The second equation in our dynamical system accounts for this, “coupling” change in the environment with the first equation. Evolutionary dynamics in this “full” model of evolution thus requires solving this system of differential equations (keeping in mind that these are a deterministic ansatz to what is ultimately an underlying set of stochastic processes).

The second equation thus specifies a function, g which describes how the environment changes over time. But notice that in neo-Darwinian evolutionary theory, according to Lewontin, we usually consider models in which environmental change is exogenous, and does not depend upon population state. Environment is external to the system of organisms and interactions being studied. We can study systems where selection is dependent upon rapidly changing, random environments, systems where selection is frequency-dependent, and systems where it is both. But we cannot, with this overall model of evolution, study systems where change in organisms depends upon the state of the population and the environment, and where change in the environment depends both upon the state of the environment and the state of the population of organisms.

And yet, the latter “reflexive” or “internalist” model is how much of the organic and cultural worlds really do evolve. We construct environments which suit us, but then we are subject to competition within those environments, which determine which folks flourish to construct the next environment we’re subject to, which define the competitive environment for the next generation, and those winners largely determine the environment, and so on….

So again, following Lewontin, a better “overall” or generic model for evolution is the following:

\begin{eqnarray}
\frac{dO}{dt} &=& f(O, E) \\
\frac{dE}{dt} &=& g(O, E) \\
\end{eqnarray}

Obviously, in the second model the function g which describes environmental change, is now fully dependent upon the state of the population. As the population evolves, it changes its environment, which leads to different dynamics in the future change of the population itself. This is “niche construction,” and when you strip it down to this level, it’s pretty apparent why some version of NCT must be true of evolving populations.

We can, of course, recover nearly any evolutionary model from this expanded ansatz. If the function g gives no, or little, weight to the parameter O, then we lose niche construction as a driver of the overall dynamical system. There are situations where we might imagine this to be the case. If we’re describing the evolution of particular traits relate only to direct solar energy flux, and the organisms have no ability to enhance or shield themselves from this flux, then there isn’t much potential for niche construction and while organismal change might still be related to both population state and environment, environmental change is fairly constant and unrelated to what organisms “do.”

The point of highlighting NCT as a major component of evolution, however, is that situations like this are rare. Most of the time, we need the full ansatz model to describe real populations and their evolution. In fact, I’d argue given the immense amount of recent work on population structure (in, say, the last decade or 15 years), that an even better ansatz is as follows:

\begin{eqnarray}
\frac{dO}{dt} &=& f(O, S, E) \\
\frac{dE}{dt} &=& g(O, S, E) \\
\frac{dS}{dt} &=& h(O, S, E) \\
\end{eqnarray}

This final ansatz, of course, points out the nearly orthogonal role that population structure plays in evolution, leading to different outcomes for any given environment or population state, depending upon the spatial and social structure of interaction. We have only to classify the hundreds of papers concerned with variations on the Prisoner’s Dilemma or Snowdrift models, to see that the same payoff matrices (i.e., the O parameter to function f) lead to different evolutionary dynamics given different spatial or topological structures to interaction. Given this, it stands to reason that niche constructions will have different fitness effects depending upon the population structure of organisms which are constructing and inhabiting those niches. Right?

I certainly think so, and I’m betting that the third ansatz model here brings together the NCT insights of Lewontin/Laland/Feldman, with the insights of Nowak and others who study evolution on complex interaction structures, to form the core of evolutionary theory for the 21st century.

## Why Studying Spatiotemporal Complex Systems Matters…

…even though it’s really tough.

And studying the full spatial behavior of stochastic processes (including evolutionary theory, in its many guises), especially when interaction and fitness are relative to a complex network of contacts or relationships, is hard. Usually so hard, that we don’t have analytic models for the full behavior of sets of stochastic processes operating on complex networks, or interacting in complex ways. We resort to simulation since the models we can solve are very simple, and few. And we seek guidance for the “average” behavior — the nonspatial global behavior of a model — in mean-field approximations. We temporarily ignore fluctuations, write deterministic mean-field equations for the dynamics, analyze those, and then add fluctuations back, in the form of simple white noise. We take the deterministic mean-field equations and derive pair-approximations or moment closures, and analyze at least the summary statistics for correlations between classes or traits we’re tracking, since we can’t analyze much else spatially. We reduce complex epidemic diffusion models to percolation problems. But mostly, we simulate.

## Will coevolutionary/adaptive network models be “easier” to understand than processes on fixed networks?

I’ve been studying statistical physics pretty hard lately, learning how to deal with many-body systems with a bunch of contributing factors to the dynamical evolution of a system. To a lesser extent, I’ve been studying the serious probability theory (interacting particle systems, stochastic processes) that go along with statistical physics. It’s caused me to ask questions about the last model I was looking at. I love it when that happens.

In a previous project on signaling theory, I looked at some of the newer literature on coevolutionary or “adaptive” network models. A coevolutionary network model is a dynamic process (for example, an evolutionary game theory model) whose interactions are localized to the structure of a mathematical graph or network. The network topology thus exerts an influence on the solution space of the game, and thus the outcomes which occur for any particular state of the population. In addition, the results of each round of the game have an effect upon the edges and nodes of the network itself, causing “rewiring” of the network and thus changes in the interaction between individuals for the next round. In the case of the costly signaling theory model I was exploring, the setup looks like this:

## The structure of mean-field transmission models

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:

$\frac{dI}{dt} = \beta (1-I) I - \gamma I$

## Moving beyond mean-field models in cultural transmission studies

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.