Consider labelings of the vertices a large regular graph by some finite set of symbols. I'll explain how fixing some average local quantity (like what fraction of edges have the same symbol on either side) can sometimes "essentially" determine all other average local quantities, and what this has to do with maximal-(sofic)-entropy measures. For some physics-inspired examples I'll say what the maximal-entropy measures are and sketch how they can be determined. For the Potts model, it turns out that the entropy is not a concave function of the entropy. This means that standard exponential families miss a range of energy values.