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I have not been following the blogs on network theory carefully. However I just saw that in Experiments in Markov chains John is doing something which looks (from a suitable distance) very like Bayesian phylogenetic analysis using a Markov Chain Monte Carlo (MCMC) technique.
There are 15 binary trees (with labelled tips) which represent the possible evolutionary relationships between 4 species. The MCMC technique uses (a deliberately designed) Markov Chain which visits these. One possible set of transitions is given by 'nearest neighbour interchanges'. I have added a picture at the bottom of the page showing the resulting graph of transitions.