Invariance, Quasi-Invariance, and Unimodularity for Random Graphs

We treat 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 a very general dynamical scheme of constructing and studying measures with a prescribed Radon–Nikodym cocycle.