  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.
