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.   

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Why we need to pay attention to the flood of work concerning spreading on networks

Until recently, my usual reaction to seeing the flood of papers on Arxiv about cooperation on complex networks was that “most people were reinventing the wheel” and hadn’t read the already vast literature on the subject. Nowak and team, as well as other folks centered more in the network science community, had already figured out the basics.

I think my former reaction on this was wrong. Nowak’s team, in particular, really has been focused less on networks, and more on the effects of graph motifs on evolutionary dynamics.  Much of Nowak’s work has been focused, from the book onward, upon how a well-mixed selection coefficient is rescaled given the local effect of graph motifs, and the consequent effect on the long-run selection force. This is brilliant and foundational stuff, and it’s paralleled by some of Keeley’s work in epidemiology on epidemic thresholds given different pair and triplet motif distributions, but it misses the full picture of network science.

It misses the “long run effects” of having both micro and mesoscale motif and ultimately, community structure.

I think that the mesoscale structure, in particular, that means that we all need to pay close attention to the flood of papers coming through Arxiv, because we’re not done yet learning all we can learn about the dynamics of spreading processes on arbitrary networks. Not by a long shot. Not to say that a significant fraction of papers aren’t partially or completely duplicative, but most will need some care to determine where they overlap, if at all.

This fact is virtually guaranteed by the fact that we have explored a tiny fraction of the space of complex networks. Mainly we’ve explored complex, heterogeneous graphs with properties that are “fairly close” to tractable. Departures from E-R graphs are controlled as much as possible so that we can make analytical progress. But we must always remember that large finite or infinite networks can have connectivity structures that go well beyond various exponential or power-law functions. Despite the generality of the Molloy-Reed configuration model, we mostly generate density functions for use in the configuration model which are relatively well behaved.

And even if we explore the parts of the network phase space relevant to biological populations,  the space of square-integrable functions which can describe stochastic processes on those networks is infintely larger than we’ve studied to date. And always will be. The space of networks and their effects on evolutionary dynamics is forever uncharted, however deeply we probe….

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Facebook, Google+, and the Crafting of the Global Social Network

(crossposted from my personal blog)

I was one of the “lucky,” who has a friend (and ex-coworker) that works for Google, and so I got an early invite to Google Plus, their attempt to take on Facebook head-on (i.e., after Facebook has achieved dominance, as opposed to the early Orkut days).

Google+ is oddly Facebook-like. This makes sense, given that FB is well-used by people of all ages in many countries. The design and interface are battle-tested (if also trivially and endlessly changeable). But there’s a key difference, and one that started me thinking about the real business that Facebook is in.

That difference is, of course, the prominence of “Circles” in Google+, and the near-absence of features in Facebook for segmenting and targeting your communications. Sure, one can create friend groups in Facebook, and then make status updates for just a friend group, but I’ll bet a lot of you either didn’t know that, or had never used it. Heck, I’ve never used it despite my expressed desire on Facebook for just such a feature. It’s nearly invisible on Facebook.

It’s central and prominent on Google+. Google wants us to *limit* and control, for ourselves, to whom we target our words and images. Twitter almost insists upon the opposite, that we speak boldly into the ether, and whomever is listening will hear, whether we know the person or not.

I’d bet that at Facebook, any feature which restricts the *volume* or *velocity* of messages that flow within the Facebook global social network are verboten, or anathema. But at the same time, Facebook positions itself as providing control and “privacy,” despite numerous well-publicized privacy issues.

Twitter largely self-organizes as a social network. Facebook, on the other hand, is *crafting* the global social network. It encourages us to accept the illusion of privacy in order to get us to friend more people, post more status, and expose our opinions and information than we would be willing to otherwise. We should not, as a result, study the Facebook social network as if it were a reflection of our real-life social networks, because the two networks are different both in topology and in weighting.

What Google+ is trying to do, and how that intent will translate into reality once it’s fully up and running, I have no idea. It is, perhaps, not entirely clear to Google themselves, since they seem to start with goals and ideas, and let data and experiment drive them toward an ultimate plan and implementation. In fact, I’ll bet the social network scientists and researchers at Google have studied the Facebook social network and its dynamics better than anybody else except Facebook’s social network scientists, and know a good deal about what makes it tick and what makes it sick.

But it’s safe to say that they’ve made a couple of bets. One is that Google is willing to accept a slightly lower velocity and average quantity of messages in the system. This is inevitable because people will restrict more highly to whom they send various status and messages if the means for doing so is prominent and core to the system’s operation. The degree to which this effect will be prominent is open to question, but the underlying inequality in rates is pretty much built in. They would make this bet if the increased loyalty they get from customers yields a better upside.

Second, they’re betting that running a more organic and self-structured social network will yield better growth than a manipulated and engineered social network. Here, I’d bet that Google analyzed growth rates from various kinds of node-addition processes, and found that Facebook is oversaturating its degree distribution and eventually will lose the desirable “near-scale-free” network properties (for propagation), and will tend toward a distribution with too many degree correlations to propagate information efficiently. That’s a complete conjecture on my part, but it’s backed by some solid science on the nature of information transfer on various network topologies.

So Google+ is starting out in a seemingly interesting direction: offering more well-integrated control over how and to whom we communicate, but with a familiar feel and design. The real question now is, will enough people come and play, so that we can figure out how well it works, what Google is *really* doing, and whether that’s good or bad for individuals.

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Anti-conformist cultural transmission observed in the wild…

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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.

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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.

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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.

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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.

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MathJax Test

Test lorem ipsum sic dolor amet:
\begin{equation}
E = \left \lbrace \{ i,j \} \in V^2_s : \alpha \Sigma_{i,j} \right \rbrace
\end{equation}

Lorem ipsum sic dolor amet

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Raise a toast to Douglas Adams…

This didn’t make Facebook’s status limit even with aggressive editing, but it is dedicated to our political system, with love and consternation.

The major problem — one of the major problems, for there are several — one of the many major problems with governing people is that of whom you get to do it; or rather of who manages to get people to let them do it to them.

To summarize: it is a well known fact that those people who most want to rule people are, ipso facto, those least suited to do it. To summarize the summary: anyone who is capable of getting themselves made President should on no account be allowed to do the job. To summarize the summary of the summary: people are a problem.

Douglas Adams, the pre-eminent social and political philosopher of our times.  Right behind Monty Python.  Then probably Jon Stewart.  With Friedrich Hayek and John Rawls taking a joint and distant fourth.

But Adams has a point.  People are the problem.  People disagree, for various and manifold reasons.  That disagreement is a problem, since it prevents us from fixing problems, and moving in whatever direction the body politic believes is good, given a strong following.

And now, in the United States, there are 350 million of us, and growing.  Do you know what the probability of us all agreeing is?

There are complicated stochastic models — interacting particle systems — which describe the full probability distribution of any combination of pairwise agreement statistics for this population (voter and contact models, see works by Thomas Liggett and Rick Durrett, in particular).  In such models, there are cases where the population will eventually reach consensus.  But the time required for the population to reach consensus is astronomically increasing with the number of people involved.  With hundreds of millions, we are guaranteed that no process which involves people talking to each other (this simplfies our exact situation, but….) will come to consensus in a population this size before the sun burns out, on average.  If we’re lucky — we end up with periods of metastability where we hover in a bounded region of state space before we wander off and “change” into something new.  When we look back, we see a “historical progression” but all it really consists of is the cumulative history of how we’ve agreed and disagreed.

Granted, this is a drastically simplified model.  In reality, we live in societies which are much more like the Potts model, or specifically, the q-state threshold Potts model described by Axelrod in his cultural polarization and cohesion simulations in the late 1990′s.  Their behavior is roughly similar at a macroscale, however, and consensus happens for a small range of parameters but a large part of the state space is coexistence of diversity, with endless wandering through the state space, especially near critical values.

In terms of political philosophy, what this means is that Montesquieu was correct with his “small republic” hypothesis, in empirical terms.  Consensus, and thus harmony on most aspects of social life, is possible with a small population, or with small numbers of attributes that define us as “us.”  As population rises, and the richness of what divides “us” from “them” rises in the Potts model, the more time we spend wandering through inconclusive regions of the state space, where we have lots of change and no stable customs, etc.

This means Madison might be wrong about his “big republic” hypothesis, at least in terms of the classical portrayal of these two thinkers and their relation to classical republican ideals.  But as we know from modern work on first and second-order social punishment, group formation, social network structure, green-beard models, and similar ways of creating ways out of the prisoner’s dilemma, we have ways of making “many overlapping small republics” out of  ”one big republic,” which means if we figure out a better way to blend our opinions — not the old state’s rights divisions, but some new way of slicing and dicing our diversity for purposes of developing a working majority, we have a chance of managing this big Madisonian republic while giving everyone the feeling of involved, empowered inclusion that really sits behind our concepts of citizenship and liberty.

And yes, this really was triggered by Douglas Adams.

Happy Towel Day!

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