Of Sand and Cities

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.)

Fossil data assembled by Jack Sepkoski and analyzed by David Raup show that over a 600-million-year period, the magnitude of extinction waves, as measured by the percentage of all marine species wiped out in a period, is related by a power law to the frequency of those waves. Big catastrophes, where high percentages of species disappear, don't appear to be qualitatively different from small upheavals–they just happen less frequently. This suggests that no special explanations are needed to cover the big extinction waves; even if no asteroids ever hit the earth, occasional giant extinction avalanches would still run through the ecology.

Power laws also appear in data that vary over time, such as the changing water level of the Nile. They crop up in the geometry of coastlines, mountains, and trees. And they show up in human affairs as Zipf's law, which says that the magnitude of system elements is related by a power law to the rank of their magnitude. For the populations of U.S. cities, the relationship is particularly simple: The population of a metro area is inversely proportional to its rank by population, so that a city of rank 100 (Shreveport, Louisiana, with 374,000 people in 1992) is about one-tenth the size of the 10th-largest city (Houston, with 3.96 million); the line wobbles a bit with the very largest cities, but the fit is still remarkable. Zipf's law also applies to the frequencies of word use in English and, though Bak doesn't mention it, to the numbers of papers published by scientists.

Bak uses the universality of these properties to argue that some general, underlying theory must be applicable to all the systems that display them: Self-organized criticality as seen in a sandpile must also be responsible for earthquakes, solar flares, river branching structures, urban population distributions, biological extinction waves, macroeconomic fluctuations, and a whole lot more. Universality may be proving too much, however, since it is hard to believe that word-use frequencies, for example, have anything to do with sandpiles. It is a little like saying that because IQ scores and missile impacts around a target both obey a Gaussian bell curve, they must share an underlying mechanism.

The universality argument is most persuasive when applied to physical systems that have direct analogues to the gradual energy input, dissipative friction, and chain-reaction possibilities of the sandpile. For example, earthquakes get their energy from gradual tectonic plate movement; the friction along fault lines resists movement; and when the stresses do produce sudden ruptures, the stress gets redistributed to other places. The metaphor seems like a pretty good one to me, and it appealed enough to at least three other research groups that they independently published articles titled "Earthquakes as a Self-Organized Critical Phenomenon." By 1995, there were over 100 articles published in support of the idea that earthquakes are a manifestation of SOC in the earth's crust.

When we come to Bak's efforts to model economic phenomena, however, the universality argument for his SOC hammer practically disappears. He presents no data showing power laws apply to business cycle fluctuations, which are the only phenomenon he attempts to model explicitly. His model has layers of vertically related firms mechanically ordering, delivering, and producing goods. Firms follow arbitrarily imposed fixed rules for inventory accumulation and exhaustion. The analogy to falling sand in this model is a series of orders for final goods that arrives each period. The system eventually self-organizes to a critical state where avalanches of production of all sizes are possible, and the avalanches follow a power law.

More interesting than the details of the model are Bak's comments about the difficulty of working on it with two economist co-authors, Michael Woodford and Jose Scheinkman. Other social scientists view economists pretty much the way I described the stereotype of theoretical physicists, only without the legitimacy of the atomic bomb to back up their arrogance. Bak seems surprised to find out that economists are actually more careful about mathematical rigor than physicists, and he seems perplexed that his sweeping criticisms of economic theory, pungently expressed but embarrassingly uninformed, were not taken to heart by his economist collaborators.

Maybe he should pick up a copy of The Self-Organizing Economy, adapted from a set of lectures given by Paul Krugman. "For whatever reason," Krugman writes, "the authors of articles and books on complexity almost never talk to serious economists or read what serious economists write; as a result, claims about the applicability of the new ideas to economics are usually coupled with statements about how economies work (and what economists know) that seem so ill-informed as to make any economist who happens to encounter them dismiss the whole enterprise. But it does not have to be that way." (Of course, Bak did talk to serious economists, but one gets the impression that he didn't listen.)

Krugman is an interesting case. His work as an elite economist injected increasing returns to scale into trade theory. That helped beget the deadly mutant spawn "industrial policy," which he has since tried to kill with vigorous popular writing. He does not suffer fools gladly, and is happy to puncture the pretensions of pseudo-thinkers like Lester Thurow or William Greider.

The book at hand, however, is, like Bak's work, a piece of serious popular science writing; the author tries to be engaging and clear but is not afraid to use a little mathematics. Krugman's exuberance in describing his work helps get the reader over the rough spots. As a set of lectures aimed at people with backgrounds in economics, it also includes some technical sections that would be hard going for the uninitiated. Fortunately, these can be skipped with little loss of meaning.

In the past few years, Krugman has gotten interested in spatial economics, which has been something of a professional backwater. He was intrigued by the failure of traditional urban economic models to generate modern polycentric cities such as Los Angeles. He also wanted to explain the uncanny regularity of Zipf's law for city populations: "We are unused to seeing regularities this exact in economics–it is so exact that I find it spooky," he says, noting that the law holds for data back in 1890 about as well as it does now.

Krugman has two principles of his own for explaining self-organization. One he calls "order from instability," which he applies to the formation of Los Angeles-style "edge cities." The other he calls "order from growth," which he uses to derive Zipf's law.

The edge city model posits that firms placed on a circle or an infinite line want to be near one another, because there are likely to be more customers and suppliers available in crowded locations, but they want to be apart from each other to reduce competition for inputs and customers. Provided that neither the attractive nor the repulsive force overwhelms the other, and that the attractive force works over a shorter range than the repulsive force, it turns out that any random initial distribution of businesses, no matter how even, will self-organize into a set of regularly spaced business centers. ("Order from instability" enters this story because a too-uniform distribution of firms turns out to be unstable, leading eventually to a regular spacing of similar-sized clumps.) Krugman sometimes calls this process "urban morphogenesis," because it is similar to certain stories of how cells in a developing embryo self-organize so that wings, legs, and eyes end up in the right places. This model has undeniable intuitive appeal; it's hard to imagine a radically different story that could better explain the pattern.

The random growth model for city populations is a variation on something economist Herbert Simon proposed many years ago. It supposes that cities add population in proportion to the population they already have, with "lumps" of new population attaching to existing "clumps." To prevent the largest cities from absorbing all population growth, it is necessary to add a probability of new cities forming. With this setup, Krugman proves that the distribution of urban populations evolves to a critical state represented by Zipf's law. The model is somewhat problematic, as Krugman freely discloses in his technical discussion, because the amount of new-city founding or population growth needed to get Zipf's law looks too big to represent historical United States data. But merely having a theory that is subject to refutation, and which confronts the remarkable regularity of Zipf's law, is a big step forward.

Krugman's general approach seems more plausible than Bak's for developing a general understanding of self-organization and complexity, for two reasons. First, he is willing to suppose that there is more than one process going on in the world, as shown by his instability and growth models. It really does seem absurd to suppose that the power law for word-use frequencies in English is generated by the same kind of process that determines earthquakes. SOC, order from instability, and Simon-style growth models appear to be independent explanations for power-law regularities. Second, Krugman starts with a more grounded understanding of the phenomena he studies, so that he knows better what features of reality are lost when he simplifies things in his models.

The statistical approach to understanding complexity seems unlikely ever to develop into a separate science that cuts across the wide range of subject matter discussed in these two books. But as the techniques and models are developed in each of the fields where they show promise, there may be a historical period of unusual cross-disciplinary conversation and understanding. Encounters between different academic tribes are rarely placid; adventurers on this trail are going to need a lot of exuberance and at least a little bit of arrogance.

Steven Postrel ([e-mail]) is an economist who teaches business strategy at the Kellogg Graduate School of Management at Northwestern University. He really isn't arrogant, although some of his students may think so.