Abstract
BackgroundMany 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 FindingsWe 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/SignificanceWe 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.
Highlights
Networks are widely used to both represent real-world systems for topological study [1] and as a substrate for modeling their dynamics [2]
Biological, social, and information networks fall into the broad class of ‘small-world’ networks [3], a middle ground between regular and random networks: they have high local clustering of elements, like regular networks, and short path lengths between elements, like random networks
The WS model begins with a ring of n nodes, each node connected to its nearest neighbors out to some range K
Summary
Networks are widely used to both represent real-world systems for topological study [1] and as a substrate for modeling their dynamics [2]. 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. 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
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