Limits of compact decorated graphs

TL;DR

This paper extends the theory of graph limits to decorated graphs with edges labeled by elements of a compact space, providing a unified framework for various graph models and their limit objects.

Contribution

It introduces a general approach to graph limits for decorated graphs, encompassing multiple models like multigraphs, weighted, and edge-colored graphs, via probability distributions on a compact space.

Findings

Limit objects are characterized by 2-variable functions with probability distributions on K.

Sampling processes are equivalent to knowing homomorphism numbers from decorated graphs.

The framework unifies various graph models under a common limit theory.

Abstract

Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an element of a compact second-countable Hausdorff space K. The "local structure" of these objects can be explored by a sampling process, which is shown to be equivalent to knowing homomorphism numbers from graphs whose edges are decorated by continuous functions on K. The model includes multigraphs with bounded edge multiplicities, graphs whose edges are weighted with real numbers from a finite interval, edge-colored graphs, and other models. In all these cases, a limit object can be defined in terms of 2-variable functions whose values are probability distributions on K.