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A UNIFIED APPROACH TO STRUCTURAL LIMITS  AND LIMITS OF GRAPHS WITH BOUNDED TREEDEPTH
, 2013
"... 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 natu ..."
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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 componentwise. 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 treedepth, motivated by their role of elementary brick these graphs play in decompositions of sparse
A UNIFIED APPROACH TO STRUCTURAL LIMITS  WITH APPLICATION TO THE STUDY OF LIMITS OF GRAPHS WITH BOUNDED TREEDEPTH
, 2013
"... 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 case ..."
Abstract
 Add to MetaCart
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 componentwise. Eventually, we consider the specific case of limits of graphs with bounded treedepth, 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 firstorder definable set