Of Sand and Cities

Steven R. Postrel from the May 1997 issue

How Nature Works: The Science of Self-Organized Criticality, by Per Bak, New York: Springer-Verlag, 212 pages, $27.00

The Self-Organizing Economy, by Paul Krugman, Cambridge, Mass.: Blackwell Publishers, 122 pages, $18.95 paper

Theoretical physicists are famous--or notorious--among their scientific colleagues for certain stereotypical traits: self-confidence shading into arrogance; a passion for reducing things to their essentials; a conviction that other fields could be easily mastered if they could afford to take the time; and an unwillingness to actually learn about those fields except under extreme duress.

Per Bak, the author of How Nature Works, is a theoretical physicist at Brookhaven National Labs who earned his reputation working on "critical phenomena associated with equilibrium phase transitions"and organic conducting materials. Judging from this book, he is a worthy representative of his profession.

Self-confidence? Consider the book's title. A passion for simple models? Bak says, "Our strategy is to strip the problem of all the flesh until we are left with the naked backbone and no further reduction is possible."Contempt for the achievements of other fields? Bak reports his opening conversational sally on meeting famous paleontologist and PBS stud Stephen Jay Gould, co-inventor and promoter of the notion of "punctuated equilibrium," which says that evolution occurs in spurts: "Wouldn't it be nice if there were a theory of punctuated equilibria?"(To which Gould replied, "Punctuated equilibria is a theory.") Unwillingness to learn about other fields? Consider this howler: "Similarly, the emphasis in economics is on prediction of stock prices and other economic indicators, since accurate predictions allow you to make money. Not much effort has been devoted to describing economic systems in an unbiased, detached way, as one would describe, say, an ant's nest."This attitude of cocky ignorance perhaps explains the unusual absence of any promotional blurbs on the book's jacket.

That said, How Nature Works is a good book, and it is good, in large part, precisely because of Bak's stereotypical physicist attitudes. In print, at least, what might seem arrogant comes across as a kind of innocent, childlike enthusiasm, a lack of concern for anything but the sheer joy of figuring things out. His ruthless simplifications of geology, evolution, and neurology pay off because, as Bak notes, his models describe behavior that is common across these domains. This universality means that trampling across others' turf is not only acceptable, but almost mandatory, if the underlying principles are to be exposed. Finally, for the most part, Bak wants the reader to grasp the basic logic of his arguments; only rarely does he try to persuade with flights of poetic language or brute intellectual authority.

Scholars from different disciplines often feel a greater sense of kinship because of shared techniques than they do because of shared subject matter. (Economic theorists, for example, are much closer in style and interests to theoretical evolutionary biologists than they are to occupational sociologists.) As a result, new disciplines that apply common techniques to a wide variety of phenomena are more attractive to the adventurous scholar than those applying disparate techniques to a narrow range of phenomena.

These days, the grandest of the "I have a hammer and look! you're really working with nails"adventures is the study of "complexity."From the Santa Fe Institute to campuses all over the world, computer simulations are being run in which simple rules applied to simple elements generate complex, ordered-looking structures. For enthusiasts of spontaneous order in human affairs--those who maintain a sense of wonder at how intricate, unplanned networks manage to provide everything from drain cleaner to symphonies--the ongoing search for principles of self-organization carries high stakes. At the very least, the prospect that such principles might be discovered should weaken knee-jerk resistance to rigorous mathematical theorizing about social life.

Much of the early work in this area was exploratory--showing how the subtle flocking behavior of birds in flight could be mimicked by simple software rules, for example-- without really developing underlying insights or models that could be confronted with data from different fields. Traditionalists could laugh about "video games"attempting to pass as science.

But now, the complexity geeks are starting to get somewhere. On one front, John Holland and his co-workers, from a math and computer science perspective, are working on tools for modeling what they call "complex adaptive systems."(See "Learning Curve," December 1996.) The other advancing front is what might be called the "statistical systems" approach. The idea here is to note certain statistical facts about the complex macroworld, posit simple processes operating among individual elements in the microworld, and then show how (and preferably why) these processes cause those statistical patterns to appear at the macro level. Stuart Kauffman, a biologist at the Santa Fe Institute, has taken this approach in his efforts to explain the origin of life and the evolution of species. (See "Who Ordered That?,"February 1996.) But Bak, the senior collaborator on many of the most important papers from this perspective, is probably the central figure.

Bak's all-purpose hammer is self-organized criticality. A system is critical if it is poised to undergo structural changes, including possible major upheavals, upon being given a small push. The criticality of a system is self-organized if it ends up in the poised state as the result of a series of "natural"small pushes.

Bak's paradigm is the sandpile. Think of dropping individual grains of sand onto a flat surface. As the grains cumulate, they start to form a pile, which is very shallow at first. During this early period the system is not critical; each new grain of sand might dislodge a few of the grains, but that's it. As the pile grows, however, its sides become steeper, and now there is the possibility of a small domino effect, where one grain's movement downhill dislodges others, which in turn dislodge still others.

For a while, these avalanches are not big enough or frequent enough to stop the gradual steepening of the sandpile. All the disturbances are local, and grains of sand far from an avalanche are not affected. But then, Bak says, "Eventually the slope reaches a certain value and cannot increase any further, because the amount of sand added is balanced on average by the amount of sand leaving the pile by falling off the edges.�It is clear that to have this average balance between the sand added to the pile, say, in the center, and the sand leaving along the edges, there must be communication throughout the entire system. There will occasionally be avalanches that span the whole pile. This is the self-organized critical (SOC) state."

Bak argues that self-organized criticality, as exemplified in the sandpile model, looks like a plausible general-purpose hammer for understanding complex systems because a wide range of them possess properties which can be derived from sandpile-type models. These properties are statistical; they do not say exactly what is going to happen next, but they do constrain the possible patterns of events that can be observed. It turns out that both the SOC sandpile and a number of other real-world systems tend to behave according to what are known as "power laws."

If the logarithm of the number of avalanches in a SOC sandpile is plotted against the logarithm of the size of the avalanches, the result is a straight line. Such a linear relationship in the logarithms is called a power law. (The name is apt because an equivalent statement is that the frequency of an avalanche of a given size is equal to that size raised to some power.) In the study of earthquakes, the line is called the Gutenberg-Richter law, and the fit of the data, judging from Bak's pictures, is remarkable. "It turns out that every time there are about 1000 earthquakes of, say, magnitude 4 on the Richter scale"he writes, "there are about 100 earthquakes of magnitude 5, 10 of magnitude 6, and so on."(The Richter scale is the logarithm of the energy released in a quake.)

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