Network ‘Small-World-Ness’: A Quantitative Method for Determining Canonical Network Equivalence

TLDR

Many technological, biological, social, and information networks are classified as “small‑world” because they exhibit tightly clustered nodes and a mean path length comparable to a random graph of the same size, yet the current categorical definition creates uncertainty about a network’s small‑world status. The authors aim to develop a precise, quantitative measure of small‑world‑ness, S, to replace the imprecise equivalence to the Watts‑Strogatz model. They define S as the trade‑off between high local clustering and short path length, declare a network small‑world if S > 1, and test this metric on a large dataset of real‑world systems. The study finds a linear relationship between S and network size n across diverse systems, demonstrates a method to assign a unique Watts‑Strogatz model to any real‑world network, and shows that this linearity arises from a common limiting growth process, thereby quantifying the notion of small‑world‑ness.

Abstract

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.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.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.