direction. “Just because I link to CNN, that doesn’t mean that CNN links to me,” Chayes said. (Although, of course, it should.)
Naturally, good network math also needs to show that the Web is scale-free. A few sites have a huge number of links, more have a medium number, and most have very few links at all. Chayes and Borgs emphasized that equations describing the Web should predict not only such a distribution of links, but also the presence of a “strongly connected component” of Web pages. In the strongly connected component, or SCC, you can move from any page to another by following hyperlinks one page at a time. If her page is in the SCC, says Chayes, she can find a path to any other SCC page. “I can follow a series of hyperlinks and get to that person’s page, and that person can follow a series of hyperlinks and get back to my page.”
Borgs pointed out that some Web pages can link into the SCC even though no path links back to them. Some pages get linked to from a page in the SCC but don’t link back to an SCC page. Knowing which pages are within the SCC, or connected to it in which way, would be important information for Web advertisers, he noted.
Building mathematical models that reproduce all these features of the Web is still a work in progress. But the models of the Web and other networks devised so far suggest that mathematicians of the future may someday be able to explain the behavior of many networks encountered in human affairs—such as economic, political and social networks; ecosystems; protein networks inside cells; and networks of contact that spread diseases. “I think there is going to be a mathematics of networks,” said Chayes. “This is a very exciting new science.”
Since game theory also claims dominion over describing human behavior, I asked Chayes whether it had any role to play in the new math of social networks. Fortunately, she said yes. “We are trying to explain why these network structures have evolved the way they have evolved, and that’s really a game theory problem,”