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.

翻译：我们针对两种具有“小世界”特性的网络模型——纽曼-沃茨模型与哈拉里模型——在等价性及技术实现方面相对鲜为人知的事实展开论述。我们提供这些模型最精确的版本（基于对原始文献的忠实还原），以澄清其与当前最流行网络分析软件包中采用的变体之间存在的偏差。这些模型在技术实现上的差异可能被视为细枝末节，但我们发现，这种可能无意造成的修正引发了显著且值得关注的变化。对于哈拉里模型，原始公式中的随机性使得聚类系数与平均最短路径长度具有更广泛的取值空间。对于纽曼-沃茨模型，由于度分布的显著差异，聚类系数同样可能受到影响，这一结论通过我们的高阶解析推导得到验证。在此过程中，我们发现了纽曼-沃茨模型（在网络科学或物理学界更为知名）与哈拉里模型（在图论或数学界更为知名）在变量奇偶性受约束的特定条件下的等价性，这将架起这两个分别在不同领域相对独立发展起来的模型之间的桥梁。我们的研究结果凸显了网络模型构建中每个细节步骤的重要性，以及模型之间可能存在的深层关联——即使它们在诞生年代或所属学科领域中最初看似截然不同。