The small-world phenomenon formalized in this article as the coinci- dence of high local clustering and short global separation, is shown to be a general feature of sparse, decentralized networks that are neither completely ordered nor completely random. Networks of this kind have received little attention, yet they appear to be widespread in the social and natural sciences, as is indicated here by three dis- tinct examples. Furthermore, small admixtures of randomness to an otherwise ordered network can have a dramatic impact on its dy- namical, as well as structural, properties—a feature illustrated by a simple model of disease transmission.
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