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We address the relatively less known facts on the equivalence and technicalrealizations surrounding two network models showing the "small-world" property,namely the Newman-Watts and the Harary models. We provide the most accurate (interms of faithfulness to the original literature) versions of these models toclarify the deviation from them existing in their variants adopted in one ofthe most popular network analysis packages. The difference in technicalrealizations of those models could be conceived as minor details, but wediscover significantly notable changes caused by the possibly inadvertentmodification. For the Harary model, the stochasticity in the originalformulation allows a much wider range of the clustering coefficient and theaverage shortest path length. For the Newman-Watts model, due to thedrastically different degree distributions, the clustering coefficient can alsobe affected, which is verified by our higher-order analytic derivation. Duringthe process, we discover the equivalence of the Newman-Watts (better known inthe network science or physics community) and the Harary (better known in thegraph theory or mathematics community) models under a specific condition ofrestricted parity in variables, which would bridge the two relativelyindependently developed models in different fields. Our result highlights theimportance of each detailed step in constructing network models and thepossibility of deeply related models, even if they might initially appeardistinct in terms of the time period or the academic disciplines from whichthey emerged.

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