Departmental Papers (ASC)

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Journal Article

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Physical Review E





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Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.


Note: At the time of this publication, Dr. Watts was affiliated with Department of Sociology at Columbia University. Currently, Dr. Duncan J. Watts is Stevens University Professor at the University of Pennsylvania, and Professor in Department of Computer and Information Science in the School of Engineering and Applied Science, Annenberg School for Communication, and Department of Operations, Information and Decisions in the Wharton School.

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Date Posted: 20 March 2023

This document has been peer reviewed.