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Not a Small World After All

Prabhakar Raghavan

Nexus: Small Worlds and the Groundbreaking Science of Networks. Mark Buchanan. 235 pp. W. W. Norton and Company, 2002. $25.95.

Some 30 years ago Stanley Milgram, a Harvard psychologist, conducted an experiment in which subjects from the Midwest were asked to convey a letter to a target in Boston. The catch was that a participant could send the letter only to someone they knew on a first-name basis, who would then do the same, with the intent of getting the letter to its final destination with the fewest such "hops." From an analysis of successful deliveries, Milgram showed that most paths used six or fewer hops, leading to the popular folklore of "six degrees of separation" between most people. Mark Buchanan's engagingly written book Nexus begins with this seminal experiment in the field of social networks and embarks on an ambitious journey illustrating connections between social networks and such diverse entities as the World Wide Web and the human brain. Buchanan draws on a variety of sources from his experience as an editor for Nature, beginning with a very readable account of the rudiments of graph theory.

He gives a nice description of the work of Duncan Watts and Steve Strogatz on the idea of "small world networks" as a basis for the phenomenon observed by Milgram and other theorists investigating social networks. Buchanan then attempts to demonstrate the applicability of the Watts-Strogatz theory of small worlds to a number of areas, such as the Internet, neural networks, food webs, the distribution of wealth and the spread of infectious diseases.

These extensions have been especially popular among physicists as a framework for addressing a variety of problems from many disciplines. Here Buchanan gets a little carried away in concluding that small worlds "work magic" and are essential to the "fabric of life," ignoring contradictory scientific evidence in some cases. For instance, he dwells on the preliminary conclusion of Albert-László Barabási and colleagues that the Web is a small world, even though subsequent detailed experimentation has shown that the Web is considerably more complex. Likewise he describes the "rich get richer" model of preferential attachment as explaining the power-law distributions of Web linkage, even though the model fails to predict the precise distributions observed in independent studies by Barabási's group and several others.

To be fair, Buchanan does occasionally use cautionary language, saying that these predictions indicate and suggest the truth rather than establish it; but his enthusiasm is apt to be misinterpreted. I believe he could have done a cleaner job of distinguishing among three kinds of evidence he presents in the book: mathematically proven facts, empirical observations and simulations on computer-generated networks. Finally, I was surprised that although Buchanan gives a fascinating account of phenomena in neuroscience, computer science, economics and epidemiology, the typical format of these anecdotes is "scientists in discipline X had a problem, and a physicist proposed a small-world explanation for this problem." Buchanan ignores the question of what the scientists from discipline X had to say before, during or after this small-world explanation. In cases of Web analysis, at least, the small-world explanations have not survived the test of time and more careful scrutiny, although this finds no mention in Buchanan's book. In some cases (for instance, the observation of power laws) there are a number of alternative explanations in the literature, some of which are more compelling from the standpoint of human behavior.

Buchanan's is one of a growing number of books on small-world networks. Certainly, any one of these is worthwhile reading for those anxious to learn about such networks. (It is misleading, though, to label the purview of this book "the science of networks"—any claim to envelop that broad discipline should include far more from the classic fields of graph theory and queuing theory.) In the process a reader gains one perspective on addressing and understanding a variety of phenomena involving fractals and self-similarity, power laws and a variety of scientific phenomena.— Prabhakar Raghavan, Computer Science–Theory, Stanford University, and Verity, Inc., Sunnyvale, California

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