  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.
