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Everyone knows the small-world phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. Duncan Watts uses this intriguing phenomenon -- colloquially called "six degrees of separation" -- as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network? The networks of this story are everywhere: the brain is a network of neurons; organizations are people networks; the global economy is a network of national economies, which are networks of markets, which are in turn networks of interacting producers and consumers. How do such networks matter? Simply put, local actions can have global consequences, and the relationship between local and global dynamics depends critically on the network's structure. Watts illustrates the subtleties of this relationship using a variety of fascinating models. This exploration will be valuable to many fields, including physics and mathematics, as well as sociology, economics, and biology.