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
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.
|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|
|Publisher:||Public Library Science|