Departmental Papers (ASC)

Document Type

Journal Article

Date of this Version

5-6-1999

Publication Source

Physical Review E

Volume

60

Start Page

7332

Last Page

7342

DOI

10.1103/PhysRevE.60.7332

Abstract

In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the cross-over from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model

Comments

Note: At the time of this publication, Dr. Watts was affiliated with Santa Fe Institute. 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.