Decomposition of the Interaction Graph of Markov Generators of Weak Coupling Limit Type

Given the interaction graph of a Markov generator of weak coupling limit type, we propose its partition by means of a discrete metric, inspired by the metric circle. This gives us simplest but richer description of its invariant states. By restricting the notions of detailed balance and non-detailed balance, we show that the complete description of the invariant state depends only on the dynamics of a complete and connected subgraph of the interaction graph.