Ne\v{s}et\v{r}il and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim is to extend the variety of constructions by considering the gadget construction. We show that, when restricting to the set of sentences, the application of gadget construction on elementarily convergent sequences yields an elementarily convergent sequence. On the other hand, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. We give several different sufficient conditions to ensure the full convergence. One of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.

翻译：Nešetřil和Ossona de Mendez最近提出了一种新的图收敛定义，称为结构化收敛。该结构化收敛框架基于一阶公式的固定片段中逻辑公式满足的概率。通过选择不同片段的灵活性，可以统一稀疏图与稠密图的经典收敛概念。由于该领域尚处于发展初期，收敛序列的已知示例范围有限，且构建方法较少。本文旨在通过引入结构构造（gadget construction）来扩展构建方法的多样性。我们证明，当限制于句子集合时，将结构构造应用于初等收敛序列会产生初等收敛序列。另一方面，我们给出反例表明，若无额外假设，无法将结论推广至完整的一阶收敛。我们提出了若干确保完全收敛的充分条件，其中一条指出：若被替换的边在原结构序列中是稠密的，则生成的新序列为一阶收敛序列。