Multigraph limits, unbounded kernels, and Banach space decorated graphs

TL;DR

This paper introduces a universal functional analytic framework for defining limit objects of Banach space decorated graph sequences, extending existing theories to multigraphs and non-compact decorations.

Contribution

It develops a generalized homomorphism density approach to graph limits, unifying various combinatorial limit notions within a Banach space framework.

Findings

Defines limit objects for Banach space decorated graphs

Extends graph limit theory to multigraph sequences

Generalizes limits for non-compact decorations

Abstract

We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions. In particular it makes it possible to assign limit objects to multigraph sequences that are convergent in the sense of node-and-edge homomorphism numbers, and it generalizes the limit theory for graph sequences with compact decorations.