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.   

Renormalization theory arises in physics whenever we seek to figure out the consequences of a detailed theory at a larger, longer, or slower scale than the theory naturally describes.  Renormalization has two origins, first in quantum field theory where it was initially used to remove infinite quantities that prevented finite calculations for the interaction of electrons and photons, and independently within statistical physics in calculating the bulk properties of materials given detailed models of molecular interaction.  Kenneth Wilson, in the early 1970′s, unified these two perspectives into the “renormalization group,” for which he later received a Nobel prize.  

The basic procedure works like this.  Imagine that we want to calculate the force between two largish atoms — say, two iron atoms.  The microscopic theory would have us add up the following quantities (at a minimum, assuming both atoms could be treated as stationary):  (a)  the forces between the two nuclei, (b) the coulomb force between each of the 52-odd electrons and the “other” nucleus, and ( c) the forces between all of the electrons themselves.  This is a complicated summation since the electrons are small and “fast” compared to the nuclei, and the force depends upon the distance between the electron and the object to which it is being compared.  So what we do in renormalization is recognize that the nuclei are slow and heavy compared to the electrons, and the “net force” between the atoms is really the force between the nuclei with a factor which represents the “average” of the forces exerted by the electrons and between the electrons.  The end result is a much simpler average formula which neglects variation on certain time and distance scales, but is hopefully accurate on larger and longer distance and time scales.  This is renormalization in a nutshell (the example, by the way, is explained in detail by Leonard Susskind in his terrific series of theoretical physics lectures available on iTunes).  

The application of this to cultural transmission theory is fairly straightforward, at least conceptually.  We tend to model the dynamics of social learning in single populations, over short-term time scales, and solve for the equilibrium states in those populations.  But we observe evidence of that learning, at least outside the laboratory or in ethnographic settings, on much longer time scales.  And often in spatially aggregated ways, such that we’re taking samples of the outcome of transmission events over whole communities or areas.  Thus, what we need to do is “integrate out” the fast and short term fluctuations and patterns, and see the longer-term swings in trait frequencies and changes in spatial patterns.  

This implies a research agenda, since this is several very different tasks.  The first is simply understanding the effects of observing cultural transmission models through the lens of temporally and spatially aggregated observations.  I made a start on this problem in a conference paper earlier this year, available in preprint form.  

The second is recognizing that we nearly always observe social learning in archaeological contexts in regional contexts, where whole communities represent a single sample of the frequencies of cultural variants.  This implies a renormalization from single population to metapopulation models, or continuous spatial models.  Some work along these lines is underway, and my own contribution is analysis of metapopulation rather than continuous diffusion models (Blythe is also working on this problem).  

The third, which is unique to my research as far as I can tell, is not treating our observable data as “snapshots” in time, which is the common pattern in archaeological CT studies.  Archaeological data, following from the first point, are temporal aggregates as is well known.  Which means that our analysis of transmission models must not compare synchronic or equilibrium predictions to diachronic observations.  We must, instead, analyze the dynamics of our models through diachronic, aggregated predictions.  This has not yet been done, at least in any literature I’ve seen.  

Finally, we observe the archaeological record through formal classifications of artifacts, and do not observe the socially transmitted cultural variation directly.  Thus, we cannot directly apply statistical distributions which are designed to apply to DNA sequences or other “more direct” measures of heritable information, to the frequencies of archaeological classes.  This problem, of course, exists within population genetics itself given the “allele/locus” models common in the pre-genomic era.  But the solution here must be uniquely archaeological given our methods of classification and observation.  We must, in other words, view cultural transmission models not through the “traits” we usually model, but analytical classifications that mimic the structure of the multidimensional types and classes we actually use.  

I do not, in this research, actually use the formal apparatus of the “renormalization group” from physics.  Exploring that apparatus and its applicability would take me far afield from the concrete and useful contributions outlined above.  But RG theory and methods are never far from my mind, and after the dissertation is completed, I want to explore its utility in more detail.  

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….

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.

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.

The significance of this problem lies entirely in the fact that most applications of cultural transmission theory employ aggregate data, and have no access to individual transmission chains or events which could be used to identify the action of different transmission processes.  This is obviously true in archaeology, where our data typically  represent not just population-level aggregates, but palimpsests of aggregates over varying amounts of time.  But it’s also true in studying contemporary populations outside the laboratory and experimental settings.

In a recent and excellent article, Steele et al. (2010) describe Bronze Age Hittite ceramic assemblages for which they have independent reasons to believe that some of the variation is not selectively neutral, but yet statistical tests cannot falsify the hypothesis of neutrality (they employ the Ewens-Watterson and Slatkin’s Exact tests).  They end up concluding that more attention is required to unit formation, refining their ideas of “what features and categories can be treated as reliable units of prehistoric cultural transmission.” (p. 1357).

My interpretation is different:  the archaeological units employed here aren’t the reason why the statistical tests are failing to detect departures from neutrality they expect to see (given vessel form and intuitive functional arguments).  What they’re possibly seeing is an instance of Mesoudi and Lycett’s open issue.

In a population where individuals may be employing different rules for learning cultural information, those rules may “add up” to a population-level statistical profile which is indistinguishable from random copying.  A simple thought-experiment example demonstrates this, although turning this post from an “open problem” to a “theorem” will require either analytic models or simulations, and possibly both.

In this illustration, we imagine individuals arrayed on a spatial lattice, interacting via transmission rules with their lattice neighbors (say, the Moore neighborhood).  Here I have pictured a situation where each individual is surrounded by Moore neighbors who have a different transmission rule than themselves (in the language of the Ising model of statistical physics, I have depicted a “maximally anti-ferromagnetic” setup).  For each individual, the effect of their biased transmission would be measurable — either as a tendency to select for imitation the most common or least common traits within their neighborhood.

But notice that as soon as we begin to aggregate trait frequencies over any non-trivial block of individuals (i.e., we examine renormalized statistics for the system), or if we examine the system-wide frequencies, the biasing forces cancel out, and we only detect “random copying.”  And not just at the level of the whole system (i.e., the macroscopic frequency spectrum), but for any intermediate block size we measure (i.e., mesoscopic frequency spectra).  We simply can’t tell, from frequency data, that the underlying microscopic transmission rules are highly biased.

Now, the situation pictured here is perfectly idealized in many ways.  In real populations, the proportions of different transmission rules will vary (often through time), and individuals will face different mixes of those rules in their social network contacts, and the strength of a conformist or anti-conformist bias might vary, depending upon the situation or the trait or the individual.  So the fact that we can construct a model in which microscopic biased transmission rules “add up” to unbiased transmission macroscopically is no guarantee that we won’t be able to detect a “signal” of bias in real populations, especially if a realistic system displays different behavior at mesoscopic scales.

Thus, the open problem here is really one of the limits and “phase portrait” of mixed systems of transmission rules.  When can we detect the mode or modes of transmission from macroscopic data?  When can’t we?  How strong do biases have to be to detect them against a background of drift, innovation, and the tendency for heterogeneity to “average out.”?  How do the answers of these questions depend upon the interaction structure (spatial or social network) of the population?

And for archaeological purposes, the answers to the questions posed in the previous paragraph must then be viewed against a background of time-averaging processes that result in archaeological assemblages.  For example, in Fraser Neiman’s (1990) dissertation, he argues that time-averaging does not obscure the statistical signature of neutral drift, and thus one can still use Wright’s F-statistics and Ewen’s tests to examine archaeological assemblages using the lens of neutral theory.  Carl Lipo’s dissertation (and our previous and subsequent work together) relies on this as well.  And it’s probably a correct assumption.  If we think about a two-trait Wright-Fisher drift model, the PDF for the trait frequency distribution is binomial (multinomial in the K-alleles or infinite-alleles case).  And if we assume that a time-averaging process is truly the process of taking the sum of counts for each time period and then recalculating the frequencies, the average of binomial (resp. multinomial) trait frequency distributions over time will be binomial (resp. multinomial) as well, and thus the time-averaged frequencies will still be distributed as expected by Ewen’s test.

We should note that time-averaging may very well have different effects when transmission is biased.  The sums of arbitrary frequency distributions may not replicate the individual frequency distributions viewed at a given moment in time.¹  Thus, we need to understand experimentally (and possibly analytically), what happens when we time-average the temporal record of different mixtures of microscopic transmission rules, before we can make any archaeological inferences.

I am hoping to work on this open problem within the next year, in the context of my dissertation research.  But I’d welcome insights or partial results if other folks have worked on aspects of this, whether in anthropology or another discipline.

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.

Random copying (also referred to by Boyd and Richerson as “unbiased transmission,” perhaps a better term) refers to an individual-level process of selecting a cultural model at random (perhaps from those individuals with whom one is geographically adjacent, or has a link within a social network model), and adopting a trait from that individual.

Drift, on the other hand, is pure sampling error which occurs whenever a stochastic process involves sampling from a finite set.  This is the sense in which Sewall Wright defined the concept in 1930.

The difference is subtle, but important.  The reason it’s important is that both processes, or only one, can be operative within a population at the same time.  We can, in other words, envision a small population within which individuals copy each other at random, and which is small enough that drift has a large “smearing out” effect on trait frequencies.

We can envision a population, in contrast, where individuals use different rules for learning cultural behavior:  perhaps transmission is biased toward the most common trait within their peer group, or biased toward emulation of individuals perceived as the most successful or powerful.  If that population were reasonably small, we would also expect sampling error (drift) to introduce additional variance into trait frequencies, beyond that created by the biased transmission rule.  In fact, we would expect that drift would lessen the “strength” of the statistical “signal” of the biased transmission rule, making it harder to measure the bias.

The latter example is the fundamental reason why we need to clean up our language (and our thinking!) and not conflate “drift” with “random copying.”  Real people will display a mix of transmission rules, perhaps depending upon the situation or kind of trait involved, and real populations of cultural models are often small enough for sampling effects to be important.

So the term “random copying” should be reserved for situations where we are explicitly proposing that individuals are not biasing selection of cultural models in an appreciable or measurable way.  And we should expect that “drift” will be a measurable force in any small population, regardless of what transmission rules are being employed by individuals.

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.

The Structure of Evolutionary Models:  Organism State, Environment, Structure

If we were to map every single interaction between organisms in a population, we’d find that different types of interactions had different patterns. We’d have one graph describing sexual relationships, another graph describing descent and relatedness, another graph describing economic “working” relationships, perhaps another graph describing “friendship,” and so on. We could describe all of these overlapping graphs as a single hypergraph. Or we can use the set-based abstraction introduced by Tarnita and Nowak to do the “bookkeeping” for how multiple kinds of complex interaction structures affect the frequencies of traits across the population. Either is probably a workable approach, but the latter is attractive for its clarity.

What this type of approach leads to is a recognition that evolution (whether genetic or cultural) is simply the name we give to various processes by which variation is sorted and filtered over time within a population, according to some general rules.  Two major theoretical developments are leading to a unified model for what those rules are and how they interact.  At the highest level, we can summarize them 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}

In the above ansatz, O is the population state (think “organisms”), S is population structure, and E is the environment. The functions f, g, and h represent a model for how organismal state, population structure, and the environment each evolve. Since each function has the same three parameters, this set of three coupled differential equations describe the co-evolution of all three simultaneously.  In other words, in the full evolutionary process, the “fitness” of a population or individual state is dependent not just upon the external environment, but upon the structure of interactions among individuals, and the distribution of individual states in the population.

Similarly, while “the environment” certainly has an exogenous element, not determined by the evolutionary process itself, there is a strong component of “felt or experienced” environments which does co-evolve with populations, and thus a full model for evolutionary processes must incorporate an understanding of this feedback loop.  This understanding goes by the general name of “niche construction theory” in contemporary evolutionary biology.

And finally, we cannot understand the fitness of individuals in their environment (as constructed endogenously as well as exogenous conditions) without understanding the structure of individual interaction.  And at the same time, we must recognize that the structure of individual interaction itself evolves as part of the processes under consideration.  The latter understanding is studied by Gross, Blasius, and others as “adaptive network models.”

This generic modeling framework or ansatz thus encompasses the traditional Modern Synthesis view of evolutionary processes, contemporary niche construction theory, game theory and frequency-dependent selection, and evolutionary “set” or “graph” theory (to use Nowak’s terminology for the effects of structure on evolution).

To the extent that we treat one or more of these functions as (a) constant, or (b) varying much more slowly than the others, we recover or approximate traditional modeling frameworks.

For example, to the extent that we treat functions g and h as approximately constant, we recover models of constant fitness and/or frequency-dependent selection — i.e., we recover adaptive dynamics and non-spatial population genetics.

To the extent that we treat just h as constant, and in fact ignore its contribution in f and g, we recover theories of selection and evolution in temporally variable environments.

Understanding “OSE” Models

Of course, when one fleshes out the functions f, g, and h to construct an actual model, we would expect the resulting model to be fairly difficult to analyze.   In general, moving beyond mean-field models drastically increases the difficulties inherent in analyzing and understanding a model. In many cases, models that incorporate frequency-dependent interactions between individuals, with a rich interaction structure, and possibly a changing environment, will not be directly solvable. Instead, such models will largely be studied using rigorous numerical simulation and Monte Carlo methods.

It also means that the methods of statistical physics are highly relevant to their analysis. Basically, it’s not just the form of the above “fitness” or “evolution” functions and their dynamics that matter, nor the probability of interaction between any set of individuals. These are simply the “mechanics” of a model which obeys the above framework.

In the above model, the evolution functions for state, environment, and structure form a probability space for evolutionary outcomes.  The probability space is critical to understand, more so than the exact behavior of the fitness and evolution functions.  The probability space for outcomes tells us “how likely” a specific outcome is, as a function of the types of environmental processes, interaction structure, and organismal state we posit within the model.

After all, we usually have a unique sequence of evolutionary changes, and what we’re seeking is the set of processes and parameter values which can account for the observed sequence of evolutionary outcomes. But we usually don’t have experimental replicates. Thus the usual methods of experimental analysis (e.g., ANOVA) aren’t easy to use, or even valid in some cases.  What we have, instead, is a multi-model inference problem of the kind discussed by Burnham and Anderson. We have a single “outcome” to use as data, and really what we want is to understand: for which models does the outcome data fit within a confidence interval of the model’s probability distribution? Furthermore, which model explains how much of the information contained in the outcome data?

In future posts, I’ll try to unpack the practicalities involved in building specific models from this general framework, incorporating the “evolutionary set theory” approach of Nowak and Tarnita, as applied to structured models of cultural transmission.

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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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!