Former *Nature *editor Buchanan (*Ubiquity*, 2001) takes an intriguing, accessible look at the mathematics behind the “six degrees of separation” theory.

In 1998, Cornell mathematician Duncan Watts was focused on a seemingly non-mathematical problem. In New Guinea, male fireflies by the millions perch on trees at night and flash their lights to attract females in perfect synchrony. With his advisor, Steve Strogatz, Watts was working on “graph problems,” a special mathematical term describing any collection of dots connected by lines. The fireflies are the dots. Their coordinated lighting indicates information transfer, which is the equivalent of connecting lines. Watts and Strogatz’s breakthrough was to see the structural similarity between the fireflies and the theory that the world’s six billion people are all connected by six degrees of separation. Degrees of separation are the number of steps needed to get from one randomly selected dot to another. Watts and Strogatz showed that when networks of connected dots have a high degree of order to their clustering, the degree of separation is correspondingly high; adding random links, however, radically shrinks the degree of separation. Networks, in other words, combine order and chaos to form “small worlds.” Subsequent chapters maneuver through Watts and Strogatz’s work as they explain the form of the Web, the food chain, epidemiology, income distribution, and many other disparate networks. By adding the evolution of the network as a second variable, Buchanan derives two basic types of small worlds: the “aristocratic,” in which the concentration of connections goes through a few “hubs”; and the egalitarian, in which connections have no particular concentration. He suggests that small-worlds theory should change the way we think about social policy.

Despite the author’s penchant for distracting digressions, a terrific, essential addition to the library of popular-science books.