national trade, protein chemistry, airline routes, and the spread of disease.

Using math to study networks is not entirely new, though. In fact, network math goes back at least to the 18th century, when the Swiss mathematician Leonhard Euler gave intellectual birth to the field with his analysis of a network of bridges in Königsberg, in eastern Prussia. In the mid-20th century, Paul Erdös and Alfréd Rényi developed the math to describe networks in their most abstract representation—essentially dots on paper connected by lines. The dots are known as nodes (or sometimes vertices); the lines are officially called edges, but are more popularly referred to as links. Such drawings of dots and lines are technically known as graphs, so traditional network math is known as graph theory.2

A graph’s dots and edges can represent almost anything in real life. The nodes may be any of various objects or entities, such as people, or companies, or computers, or nations; the links may be wires connecting machines, friendships connecting people, common film appearances connecting movie actors, or any other common property or experience. People, of course, belong to many different kinds of networks, such as networks of family, networks of friends, networks of professional collaborators. There are networks of people who share common investments, political views, or sexual partners.

Traditional graph theory math does not do a very good job of describing such networks, though. Its dots and lines resemble real networks about as much as a scorecard resembles a baseball field. The scorecard does record all the players and their positions, but you won’t get much of an idea of what baseball is like from reading the scorecard. Same with graphs. Standard graph math describes fixed networks with nodes connected at random, whereas in the real world, networks usually grow, adding new parts and new connections, while perhaps losing others—and not always at random. In a random network, every node is an equal among many, and few nodes get much more or less than a fair share of links. But in many real-world networks, some nodes possess an unusually high number of links. (In a network of sexual partners, for ex-



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