Revisiting small-world network models: Exploring technical realizations and the equivalence of the Newman-Watts and Harary models

TL;DR

This paper clarifies the technical differences and equivalence of the Newman-Watts and Harary small-world network models, revealing how subtle modifications impact their properties and bridging models from different scientific disciplines.

Contribution

It provides accurate formulations of both models, analyzes the effects of their variants, and establishes conditions under which they are equivalent, connecting network science and graph theory.

Findings

Original formulations allow wider clustering and path length variation.

Higher-order analysis shows degree distribution impacts clustering.

Models are equivalent under specific parity conditions.

Abstract

We address the relatively less known facts on the equivalence and technical realizations surrounding two network models showing the "small-world" property, namely the Newman-Watts and the Harary models. We provide the most accurate (in terms of faithfulness to the original literature) versions of these models to clarify the deviation from them existing in their variants adopted in one of the most popular network analysis packages. The difference in technical realizations of those models could be conceived as minor details, but we discover significantly notable changes caused by the possibly inadvertent modification. For the Harary model, the stochasticity in the original formulation allows a much wider range of the clustering coefficient and the average shortest path length. For the Newman-Watts model, due to the drastically different degree distributions, the clustering coefficient can also be affected, which is verified by our higher-order analytic derivation. During the process, we discover the equivalence of the Newman-Watts (better known in the network science or physics community) and the Harary (better known in the graph theory or mathematics community) models under a specific condition of restricted parity in variables, which would bridge the two relatively independently developed models in different fields. Our result highlights the importance of each detailed step in constructing network models and the possibility of deeply related models, even if they might initially appear distinct in terms of the time period or the academic disciplines from which they emerged.