(actor or airport) is as likely to be linked as much as any other, so most of them are linked to about the same degree. Only a few would have a lot more links than average, or a lot less. If actors were linked randomly, their rankings by number of links would form a bell curve, with most of them close to the middle. But in many small-world networks, there is no such “typical scale” of the number of links.
Such distributions—with no typical common size—are known as “scale free.” In scale-free networks, many lonely nodes will have hardly any connections at all, some nodes will be moderately well connected, and a few will be superconnected hubs. To mathematicians and physicists, such a scale-free distribution is a sure sign of a “power law.”
In a groundbreaking paper published in Science in 1999, Réka Albert and Albert-László Barabási of Notre Dame University noted the scale-free nature of many kinds of networks, and consequently the usefulness of power laws for describing them. The revelation that networks could be described by power laws struck a responsive chord among physicists. (They “salivate over power laws,” Strogatz says—apparently because power law discoveries in other realms of physics have won some Nobel Prizes.)
Power laws describe systems that include a very few big things and lots of little things. Cities, for example. There are a handful of U.S. cities with populations in the millions, a larger number of medium-sized cities in the 100,000 to a million range, and many, many more small towns. Same with earthquakes. There are lots of little earthquakes, too weak to notice; a fewer number of middling ones that rattle the dishes; and a very few devastating shocks that crumble bridges and buildings.
In their Science paper, Barabási and Albert showed how the probability that a node in a scale-free network is linked to a given number of other nodes diminishes as the number of links increases. That is to say, scale-free networks possess many weakly linked nodes, fewer with a moderate number of links, and a handful of monsters—like Google, Yahoo, and Amazon on the World Wide Web. Nodes with few links are common, like small earthquakes;