Abstract
In this paper we introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory.
As an outcome of this, we provide extensions of known results. We believe that this puts these into a broader context.
The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise.
We also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as "elementary bricks" these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case.
This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of modeling we introduce here.
Our example is also the first "intermediate class" with explicitly defined limit structures where the inverse problem has been solved.