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Unformatted text preview: The Small-World Phenomenon: An Algorithmic Perspective * Jon Kleinberg Abstract Long a matter of folklore, the small-world phenomenon the principle that we are all linked by short chains of acquaintances was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960s. This work was among the first to make the phenomenon quantitative, allowing people to speak of the six degrees of separation between any two people in the United States. Since then, a number of network models have been proposed as frameworks in which to study the problem analytically. One of the most refined of these models was formulated in recent work of Watts and Strogatz; their framework provided compelling evidence that the small-world phenomenon is pervasive in a range of networks arising in nature and technology, and a fundamental ingredient in the evolution of the World Wide Web. But existing models are insufficient to explain the striking algorithmic component of Milgrams original findings: that individuals using local information are collectively very effective at actually constructing short paths between two points in a social net- work. Although recently proposed network models are rich in short paths, we prove that no decentralized algorithm, operating with local information only, can construct short paths in these networks with non-negligible probability. We then define an infi- nite family of network models that naturally generalizes the Watts-Strogatz model, and show that for one of these models, there is a decentralized algorithm capable of finding short paths with high probability. More generally, we provide a strong characterization of this family of network models, showing that there is in fact a unique model within the family for which decentralized algorithms are effective. * A version of this work appears as Cornell Computer Science Technical Report 99-1776 (October 1999). Department of Computer Science, Cornell University, Ithaca NY 14853. Email: email@example.com. Supported in part by a David and Lucile Packard Foundation Fellowship, an Alfred P. Sloan Research Fellowship, an ONR Young Investigator Award, and NSF Faculty Early Career Development Award CCR- 9701399. 1 1 Introduction The Small-World Phenomenon. A social network exhibits the small-world phenomenon if, roughly speaking, any two individuals in the network are likely to be connected through a short sequence of intermediate acquaintances. This has long been the subject of anecdotal observation and folklore; often we meet a stranger and discover that we have an acquaintance in common. It has since grown into a significant area of study in the social sciences, in large part through a series of striking experiments conducted by Stanley Milgram and his co-workers in the 1960s [13, 18, 12]. Recent work has suggested that the phenomenon is pervasive in networks arising in nature and technology, and a fundamental ingredient in the...
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This note was uploaded on 04/20/2008 for the course STAT 260 taught by Professor Davidaldous during the Spring '07 term at University of California, Berkeley.
- Spring '07