Tag evolutionary modeling

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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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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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.

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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:

adaptive-network-model.png

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