Ubiquity

The science of history … or why the world is simpler than we think.

by Mark Buchanan

283 pages, ISBN 0-609-60810-X, Crown, New York, 2001.

www.randomhouse.com

Reviewed by J. M. Haile, Macatea Productions, http://www.macatea.com/

Our everyday experience teaches us that, in many common situations, small effects have small causes while big effects have big causes: if we gently push a wagon that sits on a level pavement, it moves just a little, but if we give it a hard shove, it moves much farther. Similarly, if we barely depress the accelerator on an automobile, the car moves slowly, but if we put pedal to the metal, the vehicle moves quickly indeed. Of course, we also learn about exceptions; for example, if a wagon is poised at the top of a hill, then only a gentle push is needed to send it rolling a long way. But usually such exceptions are easily recognized, and they don't distract us from the general rule: the size of the effect is proportional to the size of the cause.

Given this experience, it is natural for us to expect that a "big" event can only be understood and explained by finding a proportionally "big" cause. Examples of "big" events include a magnitude 8 earthquake, a forest fire that burns a million acres, a crash in stock markets, the extinction of the dinosaurs, a world war. A crucial test for understanding is whether an explanation can be used to make a reliable prediction: we believe our explanations about the dynamics of the solar system because they enable us to predict the phases of the moon, solar and lunar eclipses, transits of the planets, and a wealth of other detailed behavior. In contrast, no geologist has ever successfully predicted the occurrence of a single large earthquake; no economist has successfully predicted when stock markets will crash; historians cannot even agree on the causes of previous world wars, much less predict when or whether another will occur.

The extinction of the dinosaurs is particularly instructive: The "big" cause is nowadays believed to be the crash of a large meteorite into the earth, and, indeed, an appropriately sized and dated crater has been located off the Yucatán Peninsula. But in the history of life on earth, other mass extinctions have occurred, some bigger than that involving the dinosaurs, and no meteoric crater has been linked to those events. Further, other large craters have been found on earth to which no mass extinctions can be linked. So while it might be that a meteorite triggered the extinction of the dinosaurs, it appears that a large meteorite is neither a necessary nor a sufficient explanation.

The thesis of Mark Buchanan's book is that there are dynamic systems, including those just introduced, in which big events do not have big causes; instead, big events in such systems are initiated by the same small causes as small events. Such systems are composed of a large number of dynamic elements that are linked into complex networks through nonlinear relations. For earthquakes, the elements are rocks, linked into networks to form tectonic plates, with the dynamics provided by plates sliding past one another. For stock markets, the elements are individual traders, linked into a marketing network, with the dynamics realized through buying and selling. For mass extinctions, the elements are individual species, linked to other species through food chains and competition for environmental resources.

In these dynamic systems, small events occur routinely. Along the San Andreas fault in California, the US Geological Survey records an earthquake almost every hour, but the huge majority of these are small—less than magnitude 3. (You can see current seismographic recordings for California at the USGS website.) For stock markets, any reliable index, such as Dow-Jones or Standard and Poor's, exhibits fluctuations over time, but the huge majority of the down-ticks in those fluctuations are small compared to a "crash". Over the history of life on earth, individual species have routinely become extinct, but usually only a few go extinct within any given period.

The idea here is that, in any one of such systems, large events occur for the same reasons and via the same mechanisms as small events: large events are merely small events magnified in scale. To illustrate, consider this game. We take a few hundred dominoes and stand them on a table top, each domino stands on one of its ends. We start an "event" by randomly selecting one domino and pushing it over. This first falling domino may push over a second, creating a domino effect. After an event, we set up the fallen dominoes in new randomly chosen positions. We play this game over tens of thousands of events, from which we intend to obtain the probability distribution P(N) that N dominoes fall in one event. This would allow us to determine, for example, the most probable value for N.

There are two important parameters that influence the behavior revealed by this game: the dimensions of each domino (all have the same height, width, and thickness) and the size of the table top. By playing the game, we learn how to adjust these parameters so that the probability distribution P(N) follows a power law with negative slope. This means that in most events, only a few dominoes fall; however, events can occasionally occur in which many, many dominoes are knocked over. In any one event the number that fall is determined only by which domino is pushed over first and on how all the dominoes stand in relation to one another. But no matter whether only two dominoes fall or 200 fall, every event has the same initial cause (one domino is pushed over) and every event has the same basic structure (a falling domino knocks over a neighbor). The initial event that caused N = 200 is no different from the cause for N = 2; that is, a big event does not have a special, unique, extraordinary cause. Every event is a chain reaction, but a few chains are much longer than most others, simply because of the ways the dominoes become organized with respect to one another.

The argument is that the behavior in the domino game can also be found in other natural and man-made systems, including earthquakes, forest fires, stock markets, and species extinctions. In each case, the telltale signature is that the "size" or "strength" of events has a power-law probability distribution. A power law implies scale-invariance: events have the same structure on all scales, so a big event is merely an enlarged small event. Since the size of an event is determined by the ways in which system elements are organized with respect to one another, and since the organization changes as the dynamics of the system evolves, we can say that the behavior of the system during an event is influenced by the system's history. In the domino game, the lengths of various chains over the table are determined by the history of the game to that point; only one of those chains will actually be activated, however, depending on which domino is randomly selected to topple first.

The behavior we have just described explains why it is essentially impossible to make predictions about future events in these systems. Let's recall the two kinds of dynamic situations in which we can make predictions. One is when the system involves only a few objects moving under known forces. In these cases, motion is described by differential equations, and if we have a set of initial conditions for applied forces, then we can solve the differential equations to compute the motion. For example, we know the differential equation that describes the motion of a tennis ball in the earth's gravitational field. So if we know the strength and direction of the force with which we hit the ball, then we can compute its trajectory and predict where the ball will hit the earth. But knowledge of the initial forces applied to a dynamic network of interrelated objects may not be sufficient for us to predict the size or strength of the event that follows. An earthquake starts when one rock slips, causing another to slip, and hence setting up a chain reaction. But even if we could identify the forces that cause the first rock to slip, this would not be sufficient for us to determine the magnitude of the ensuing earthquake. We would also need to know the history of the fault line because history determines how that first rock is enmeshed within the forces acting through chains of rocks distributed throughout the tectonic plate. Given the organizational complexity of rocks along a geological fault line, that knowledge appears to be beyond our grasp.

Predictions are also possible when dynamic systems are at equilibrium. In these cases, most events are characterized by a probability that obeys the familiar Gaussian distribution. Recall a Gaussian for P(N) has a maximum which represents both the average value for N and its most probable value. Consequently, we expect most events to have strengths near the average. But the systems under consideration here, such as those leading to earthquakes and mass extinctions, are not at equilibrium and the distributions obey a power law, not a Gaussian. A power law is monotonic, and the average value for N is not the most probable value. In fact, the most probable N and the average N may differ considerably. In a real sense, there is no typical value for N: most earthquakes are weak, but the "average" quake is not so weak, and destructive quakes cannot be ruled out. Since the average N fails to coincide with the most probable N, the average does not offer a fruitful basis for making predictions.

The ideas presented in this book are collectively referred to either as critical-state universality or as complexity theory. Buchanan prefers the former, but in any case, he uses the first half of the book to explain and illustrate the ideas. His presentation is excellent: technically accurate, dramatically compelling, very readable. He does a fine job of distinguishing among what is definitely known, what is probably correct, and what is guess work. Interestingly, he often introduces mistakes made in early research and then he describes how those mistakes were recognized and corrected in later work. Since the dynamic evolutions of these systems create histories, Buchanan, in the second half of his book, speculates on the extent to which these ideas can be extended to the histories and behaviors of human social structures. Again, he is careful to emphasize that much of what is presented in these last chapters is logically extended speculation, which nevertheless might motivate a more scientific and mathematical approach to the study of human affairs.

This book was completed before 9/11, and so it cannot deal with the use and abuse of complexity theory in the planning, executing, and interpreting of the American presence in Iraq. There is a growing literature on this subject, much of it entailing misinterpretations of the theory twisted to fit a writer's own political agenda. But if you seek an politically unbiased introduction to complexity theory, one that clearly uncovers the scientific merits and limitations of the ideas, Buchanan's book is definitely the place to start.

(jmh 13 July 06) © 2006 by J. M. Haile. All rights reserved.