Invariance, quasi-invariance and unimodularity for random graphs

TL;DR

This paper connects unimodularity in random graphs to measured equivalence relations, showing that unimodular measures are a special case of quasi-invariant measures characterized by a modular cocycle.

Contribution

It introduces a new framework linking unimodularity to quasi-invariance and modular cocycles within the theory of measured equivalence relations.

Findings

Unimodular measures correspond to quasi-invariant measures with a specific Radon-Nikodym cocycle.

The modular cocycle provides a natural way to characterize unimodularity.

Embedding unimodularity into the dynamical scheme of Radon-Nikodym cocycles.

Abstract

We interpret the probabilistic notion of unimodularity for measures on the space of rooted locally finite connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in considering quasi-invariant (rather than just invariant) measures with respect to the root moving equivalence relation. We define a natural modular cocycle of this equivalence relation, and show that unimodular measures are precisely those quasi-invariant measures whose Radon--Nikodym cocycle coincides with the modular cocycle. This embeds the notion of unimodularity into the very general dynamical scheme of constructing and studying measures with a prescribed Radon--Nikodym cocycle.