A unified approach to structural limits, and limits of graphs with bounded tree-depth

Abstract

In this paper we introduce a general framework for the study of limits of relational structures in general 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 put these into next context and perspective. For example, we prove that the sparse--dense dichotomy exactly corresponds to random free graphons. 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 limits 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 of elementary brick 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 {\em modeling} we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures.

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References in corpus (7)

• Graph limits and exchangeable random graphs
• A Model Theory Approach to Structural Limits
• Limits, Regularity and Removal for Finite Structures
• First-order limits, an analytical perspective
• Existence of Modeling Limits for Sequences of Sparse Structures
• Limits of Structures and the Example of Tree-Semilattices
• First order convergence and roots

Cited by in corpus (11)

• Subgraph densities in a surface
• First-order limits, an analytical perspective
• The independence number of HH-homogeneous graphs and a classification of MB-homogeneous graphs
• Existence of Modeling Limits for Sequences of Sparse Structures
• Limits of Structures and the Example of Tree-Semilattices
• First order convergence and roots
• A duality theoretic view on limits of finite structures
• First order limits of sparse graphs: Plane trees and path-width
• Measures on the square as sparse graph limits
• Structural convergence and algebraic roots
• A duality theoretic view on limits of finite structures: Extended version