Chapter 1 Complexity Science
1.1 What is this book about?
This book is about data structures and algorithms, intermediate programming in Python, computational modeling and the philosophy of science:
This book is also about complexity science, which is an interdisciplinary field—at the intersection of mathematics, computer science and natural science—that focuses on discrete models of physical systems. In particular, it focuses on complex systems, which are systems with many interacting components.
Complex systems include networks and graphs, cellular automata, agent-based models and swarms, fractals and self-organizing systems, chaotic systems and cybernetic systems. These terms might not mean much to you at this point. We will get to them soon, but you can get a preview at http://en.wikipedia.org/wiki/Complex_systems.
1.2 A new kind of science
In 2002 Stephen Wolfram published A New Kind of Science where he presents his and others’ work on cellular automata and describes a scientific approach to the study of computational systems. We’ll get back to Wolfram in Chapter 6, but I want to borrow his title for something a little broader.
I think complexity is a “new kind of science” 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. Voilà!
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 (a copy is available from http://statistics.berkeley.edu/~aldous/157/Papers/Schelling_Seg_Models.pdf).
Here is a summary of the paper (from Chapter 10):
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? Most 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 complex than planets, but when you think about it, planets are just as complex as people (especially the ones that have people).
Both systems are complex, 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.
1.3 Paradigm shift?
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 law-based, expressed in the form of equations, and solved by mathematical derivation. Models that fall under the umbrella of complexity are often rule-based, 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 10. Explanations of these phenomena often use agent-based 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 disagree about what questions to ask and what tools to use to answer them. These decisions are based on value judgments, so there is no reason to expect agreement.
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.
1.4 The axes of scientific models
I 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 four-color 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 computer-assisted 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 more skepticism, and models that include imperfect information and bounded rationality are hot topics.
1.5 A new kind of model
1.6 A new kind of engineering
1.7 A new kind of thinking
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 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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