Ising model with dynamic random lattice

Question

Is there anything known about Ising models [1] in which the a site's neighbors change over time? For an extreme case, suppose the graph is held fixed but we randomly permute the states, assigning them randomly to sites. What aspects of the standard analysis survive?

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[1] I'm interested in them for this game-theoretic application -- http://arxiv.org/abs/0812.0198 -- but I think that shouldn't matter very much.