Related papers: A unified approach to structural limits, and limit…
In this paper we present a novel approach to graph (and structural) limits based on model theory and analysis. The role of Stone and Gelfand dualities is displayed prominently and leads to a general theory, which we believe is naturally…
Limits of graphs were initiated recently in the two extreme contexts of dense and bounded degree graphs. This led to elegant limiting structures called graphons and graphings. These approach have been unified and generalized by authors in a…
The goal of this paper is to unify two lines in a particular area of graph limits. First, we generalize and provide unified treatment of various graph limit concepts by means of a combination of model theory and analysis. Then, as an…
The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as "flag algebras", while the semantic, geometric one is often…
We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…
The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth. In this paper we survey results related to this concept…
We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by…
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs…
Graphons have traditionally served as limit objects for dense graph sequences, with the cut distance serving as the metric for convergence. However, sparse graph sequences converge to the trivial graphon under the conventional definition of…
We investigate structural properties of large, sparse random graphs through the lens of "sampling convergence" (Borgs et. al. (2017)). Sampling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a…
We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…
Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying…
A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel.…
In this paper we describe a triple correspondence between graph limits, information theory and group theory. We put forward a new graph limit concept called log-convergence that is closely connected to dense graph limits but its main…
We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs -- more precisely, from classes of bounded expansion and from nowhere dense classes. In both cases, the…
Nesetril and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no…
Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay…
We study limits of convergent sequences of string graphs, that is, graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We…
Can graph neural networks generalize to graphs that are different from the graphs they were trained on, e.g., in size? In this work, we study this question from a theoretical perspective. While recent work established such transferability…
In this paper, we exploit the theory of dense graph limits to provide a new framework to study the stability of graph partitioning methods, which we call structural consistency. Both stability under perturbation as well as asymptotic…