A unified approach to structural limits (with application to the study of limits of graphs with bounded tree-depth)

Jaroslav Nesetril (1) , Patrice Ossona de Mendez (1, 2)

1 IUUK - Computer Science Institute of Charles University [Prague]
2 CAMS - Centre d'Analyse et de Mathématique sociales

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 model theory and 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 our theory, we provide extensions of known results and identify some new cases exhibiting specific properties suggesting that their study could be more accessible than the full general case. The second part of the paper is devoted to the study of such a case, namely limits of graphs (and structures) with bounded diameter connected components. We prove that in this case the convergence can be ''almost'' studied component-wise. Eventually, we consider the specific case of limits 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 "modeling" we introduce here. It is also the first ''intermediate class'' with explicitly defined limit structures.

Mots clés: Model theory; First-order logic; Measurable graph; Graph; Relational structure; Graph limits; Structural limits; Radon measures; Stone space

Domaines: Combinatoire [math.CO]

Jaroslav Nesetril, Patrice Ossona de Mendez. A unified approach to structural limits (with application to the study of limits of graphs with bounded tree-depth). 2013. ⟨hal-00804823v1⟩