Gadget construction and structural convergence

TL;DR

This paper explores the gadget construction method within the framework of structural graph convergence, showing how it can generate new convergent sequences and identifying conditions for full first-order convergence.

Contribution

It extends the known constructions of convergent graph sequences by applying gadget construction, and provides conditions under which full first-order convergence is achieved.

Findings

Gadget construction preserves elementarily convergence.

Counterexamples show limitations for full first-order convergence.

Density of replaced edges ensures full convergence.

Abstract

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.