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Network 'small-world-ness': a quantitative method for determining canonical network equivalence

Humphries, M.D. and Gurney, K. (2008) Network 'small-world-ness': a quantitative method for determining canonical network equivalence. Plos One, 3 (4). Art No.e0002051. ISSN 1932-6203

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Abstract

Background: Many technological, biological, social, and information networks fall into the broad class of 'small-world' networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction ('small/not-small') rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model-the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified.

Methodology/Principal Findings: We defined a precise measure of 'small-world-ness' S based on the trade off between high local clustering and short path length. A network is now deemed a 'small-world' if S. 1-an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process.

Conclusions/Significance: We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.

Item Type: Article
Copyright, Publisher and Additional Information: © 2008 Humphries, Gurney. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > Department of Psychology (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 20 Jan 2010 11:19
Last Modified: 09 Jun 2014 08:45
Published Version: http://dx.doi.org/10.1371/journal.pone.0002051
Status: Published
Publisher: Public Library Science
Refereed: Yes
Identification Number: 10.1371/journal.pone.0002051
URI: http://eprints.whiterose.ac.uk/id/eprint/10308

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