This HTML version of Think Complexity, 2nd Edition is provided for convenience, but it is not the best format of the book. In particular, some of the symbols are not rendered correctly. You might prefer to read the PDF version. Chapter 1 Complexity ScienceThe thesis of this book is that complexity science is a “new kind of science”, a phrase I’m borrowing from Stephen Wolfram. In 2002 Wolfram published A New Kind of Science, where he presents his and others’ work on cellular automatons and describes a scientific approach to the study of computational systems. We’ll get back to Wolfram in Chapter ??, but for now I want to use his title for something a little broader. I think complexity is new not because it applies the tools of science to a new subject, but because it uses different tools, allows different kinds of work, and ultimately changes what we mean by “science”. To demonstrate the difference, I’ll start with an example of classical science: suppose someone asked you why planetary orbits are elliptical. You might invoke Newton’s law of universal gravitation and use it to write a differential equation that describes planetary motion. Then you could solve the differential equation and show that the solution is an ellipse. QED! Most people find this kind of explanation satisfying. It includes a mathematical derivation—so it has some of the rigor of a proof—and it explains a specific observation, elliptical orbits, by appealing to a general principle, gravitation. Let me contrast that with a different kind of explanation. Suppose you move to a city like Detroit that is racially segregated, and you want to know why it’s like that. If you do some research, you might find a paper by Thomas Schelling called “Dynamic Models of Segregation”, which proposes a simple model of racial segregation: Here is my description of the model, from Chapter ??: The Schelling model of the city is an array of cells where each cell represents a house. The houses are occupied by two kinds of “agents”, labeled red and blue, in roughly equal numbers. About 10% of the houses are empty. If you start with a simulated city that is entirely unsegregated and run the model for a short time, clusters of similar agents appear. As time passes, the clusters grow and coalesce until there are a small number of large clusters and most agents live in homogeneous neighborhoods. The degree of segregation in the model is surprising, and it suggests an explanation of segregation in real cities. Maybe Detroit is segregated because people prefer not to be greatly outnumbered and will move if the composition of their neighborhoods makes them unhappy. Is this explanation satisfying in the same way as the explanation of planetary motion? Many people would say not, but why? Most obviously, the Schelling model is highly abstract, which is to say not realistic. It is tempting to say that people are more complicated than planets, but when you think about it, planets are just as complicated as people (especially the ones that have people). Both systems are complicated, and both models are based on simplifications; for example, in the model of planetary motion we include forces between the planet and its sun, and ignore interactions between planets. The important difference is that, for planetary motion, we can defend the model by showing that the forces we ignore are smaller than the ones we include. And we can extend the model to include other interactions and show that the effect is small. For Schelling’s model it is harder to justify the simplifications. To make matters worse, Schelling’s model doesn’t appeal to any physical laws, and it uses only simple computation, not mathematical derivation. Models like Schelling’s don’t look like classical science, and many people find them less compelling, at least at first. But as I will try to demonstrate, these models do useful work, including prediction, explanation, and design. One of the goals of this book is to explain how. 1.1 Paradigm shift?When I describe this book to people, I am often asked if this new kind of science is a paradigm shift. I don’t think so, and here’s why. Thomas Kuhn introduced the term “paradigm shift” in The Structure of Scientific Revolutions in 1962. It refers to a process in the history of science where the basic assumptions of a field change, or where one theory is replaced by another. He presents as examples the Copernican revolution, the displacement of phlogiston by the oxygen model of combustion, and the emergence of relativity. The development of complexity science is not the replacement of an older model, but (in my opinion) a gradual shift in the criteria models are judged by, and in the kinds of models that are considered acceptable. For example, classical models tend to be lawbased, expressed in the form of equations, and solved by mathematical derivation. Models that fall under the umbrella of complexity are often rulebased, expressed as computations, and simulated rather than analyzed. Not everyone finds these models satisfactory. For example, in Sync, Steven Strogatz writes about his model of spontaneous synchronization in some species of fireflies. He presents a simulation that demonstrates the phenomenon, but then writes: I repeated the simulation dozens of times, for other random initial conditions and for other numbers of oscillators. Sync every time. [...] The challenge now was to prove it. Only an ironclad proof would demonstrate, in a way that no computer ever could, that sync was inevitable; and the best kind of proof would clarify why it was inevitable. Strogatz is a mathematician, so his enthusiasm for proofs is understandable, but his proof doesn’t address what is, to me, the most interesting part the phenomenon. In order to prove that “sync was inevitable”, Strogatz makes several simplifying assumptions, in particular that each firefly can see all the others. In my opinion, it is more interesting to explain how an entire valley of fireflies can synchronize despite the fact that they cannot all see each other. How this kind of global behavior emerges from local interactions is the subject of Chapter ??. Explanations of these phenomena often use agentbased models, which explore (in ways that would be difficult or impossible with mathematical analysis) the conditions that allow or prevent synchronization. I am a computer scientist, so my enthusiasm for computational models is probably no surprise. I don’t mean to say that Strogatz is wrong, but rather that people have different opinions about what questions to ask and what tools to use to answer them. These opinions are based on value judgments, so there is no reason to expect agreement. Nevertheless, there is rough consensus among scientists about which models are considered good science, and which others are fringe science, pseudoscience, or not science at all. I claim, and this is a central thesis of this book, that the criteria this consensus is based on change over time, and that the emergence of complexity science reflects a gradual shift in these criteria. 1.2 The axes of scientific modelsI have described classical models as based on physical laws, expressed in the form of equations, and solved by mathematical analysis; conversely, models of complexity systems are often based on simple rules and implemented as computations. We can think of this trend as a shift over time along two axes:
The new kind of science is different in several other ways. I present them here so you know what’s coming, but some of them might not make sense until you have seen the examples later in the book.
These are generalizations, so we should not take them too seriously. And I don’t mean to deprecate classical science. A more complicated model is not necessarily better; in fact, it is usually worse. Also, I don’t mean to say that these changes are abrupt or complete. Rather, there is a gradual migration in the frontier of what is considered acceptable, respectable work. Some tools that used to be regarded with suspicion are now common, and some models that were widely accepted are now regarded with scrutiny. For example, when Appel and Haken proved the fourcolor theorem in 1976, they used a computer to enumerate 1,936 special cases that were, in some sense, lemmas of their proof. At the time, many mathematicians did not consider the theorem truly proved. Now computerassisted proofs are common and generally (but not universally) accepted. Conversely, a substantial body of economic analysis is based on a model of human behavior called “Economic man”, or, with tongue in cheek, Homo economicus. Research based on this model was highly regarded for several decades, especially if it involved mathematical virtuosity. More recently, this model is treated with skepticism, and models that include imperfect information and bounded rationality are hot topics. 1.3 A new kind of modelComplex models are often appropriate for different purposes and interpretations:
We get back to explanatory models in Chapter ??, instrumentalism in Chapter ??, and holism in Chapter ??. 1.4 A new kind of engineeringI have been talking about complex systems in the context of science, but complexity is also a cause, and effect, of changes in engineering and the organization of social systems:
1.5 A new kind of thinkingWe are getting farther afield now, but the shifts I am postulating in the criteria of scientific modeling are related to 20th Century developments in logic and epistemology.
These trends are not universal or complete, but the center of opinion is shifting along these axes. As evidence, consider the reaction to Thomas Kuhn’s The Structure of Scientific Revolutions, which was reviled when it was published and now considered almost uncontroversial. These trends are both cause and effect of complexity science. For example, highly abstracted models are more acceptable now because of the diminished expectation that there should be a unique, correct model for every system. Conversely, developments in complex systems challenge determinism and the related concept of physical law. This chapter is an overview of the themes coming up in the book, but not all of it will make sense before you see the examples. When you get to the end of the book, you might find it helpful to read this chapter again.

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