Related papers: On Local Limits of Sparse Random Graphs: Color Con…

The classical Weisfeiler-Leman algorithm aka color refinement is fundamental for graph learning with kernels and neural networks. Originally developed for graph isomorphism testing, the algorithm iteratively refines vertex colors. On many…

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…

The color refinement algorithm is mainly known as a heuristic method for graph isomorphism testing. It has surprising but natural characterizations in terms of, for example, homomorphism counts from trees and solutions to a system of linear…

The Colour Refinement algorithm is a classical procedure to detect symmetries in graphs, whose most prominent application is in graph-isomorphism tests. The algorithm and its generalisation, the Weisfeiler-Leman algorithm, evaluate local…

The colored neighborhood metric for sparse graphs was introduced by Bollob\'as and Riordan. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm. We prove that even in this refined sense, the…

We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a…

Color Refinement, also known as Naive Vertex Classification, is a classical method to distinguish graphs by iteratively computing a coloring of their vertices. While it is mainly used as an imperfect way to test for isomorphism, the…

In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that…

Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global…

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…

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised…

Message-passing graph neural networks (MPGNNs) are commonly compared with the Weisfeiler-Lehman (WL) color-refinement procedure, but this comparison does not quantify the resource parameters a network needs to realize color refinement with…

Automatic color enhancement is aimed to adaptively adjust photos to expected styles and tones. For current learned methods in this field, global harmonious perception and local details are hard to be well-considered in a single model…

We precisely characterize the expressivity of computable Recurrent Graph Neural Networks (recurrent GNNs). We prove that recurrent GNNs with finite-precision parameters, sum aggregation, and ReLU activation, can compute any graph algorithm…

Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behavior and fitting properties are still critical challenges, that in general, require model specific…

Let $G=G(n,m)$ be a random graph whose average degree $d=2m/n$ is below the $k$-colorability threshold. If we sample a $k$-coloring $\sigma$ of $G$ uniformly at random, what can we say about the correlations between the colors assigned to…

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we…

We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global…

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…

Local convergence techniques have become a key methodology to study sparse random graphs. However, convergence of many random graph properties does not directly follow from local convergence. A notable, and important, such random graph…