As stated in the title I am struggling to really grasp what unimodular graphs are.
My questions stems from trying to understand this paper https://arxiv.org/pdf/1308.3755.pdf where the definition of unimodularity of the graph comes from the unimodularity of an associated measure. Which does not help with intuition. In papers written by Russell Lyons, https://arxiv.org/pdf/math/0603062.pdf I've seen unimodularity described as equivalent to the mass transport principle of transitive graphs, or by the right-invariance of the left Haar measure of Aut(G). Overall, in most papers, I found the notion of unimodularity was so "obvious" that the authors do not describe it in sufficient depth for me to learn.
I had the understanding that it relates to some notion of local invariance and self-similarity, but examples of non-unimodular graphs (the Trofimov grandparent graph for example) disprove that.
Is there an intuitive way to see unimodularity, or is it a purely calculatory property?