Subgraph densities in Markov spaces

TL;DR

This paper extends the concept of subgraph densities from dense graph limit theory to Markov spaces, defining homomorphism measures and exploring their existence under certain conditions, advancing the understanding of graph limits.

Contribution

It introduces a generalization of subgraph densities to Markov spaces and establishes conditions for the existence of homomorphism measures, broadening graph limit theory.

Findings

Homomorphism measures can be constructed under smoothness and sparseness conditions.

The generalization connects dense graph limits to Markov space frameworks.

Provides a foundation for analyzing graph sequences via Markov spaces.

Abstract

We generalize subgraph densities, arising in dense graph limit theory, to Markov spaces (symmetric measures on the square of a standard Borel space). More generally, we define an analogue of the set of homomorphisms in the form of a measure on maps of a finite graph into a Markov space. The existence of such homomorphism measures is not always guaranteed, but can be established under rather natural smoothness conditions on the Markov space and sparseness conditions on the graph. This continues a direction in graph limit theory in which such measures are viewed as limits of graph sequences.