The ubiquity of small-world networks

TL;DR

This paper critiques the traditional methods of identifying small-world networks, introduces a new metric {} to improve accuracy, and demonstrates its effectiveness with example networks, impacting network science analysis.

Contribution

The paper proposes a new small-world metric {} that better distinguishes small-world networks by comparing clustering to a lattice and path length to a random network.

Findings

Traditional clustering comparison can lead to false positives.

The new {} metric more accurately identifies small-world properties.

Some networks previously classified as small-world are not according to {}.

Abstract

Small-world networks by Watts and Strogatz are a class of networks that are highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. These characteristics result in networks with unique properties of regional specialization with efficient information transfer. Social networks are intuitive examples of this organization with cliques or clusters of friends being interconnected, but each person is really only 5-6 people away from anyone else. While this qualitative definition has prevailed in network science theory, in application, the standard quantitative application is to compare path length (a surrogate measure of distributed processing) and clustering (a surrogate measure of regional specialization) to an equivalent random network. It is demonstrated here that comparing network clustering to that of a random network can result in aberrant findings and networks once thought to exhibit small-world properties may not. We propose a new small-world metric, {\omega} (omega), which compares network clustering to an equivalent lattice network and path length to a random network, as Watts and Strogatz originally described. Example networks are presented that would be interpreted as small-world when clustering is compared to a random network but are not small-world according to {\omega}. These findings have significant implications in network science as small-world networks have unique topological properties, and it is critical to accurately distinguish them from networks without simultaneous high clustering and low path length.