Monte Carlo Markov chains constrained on graphs for a target with disconnected support

This paper presents a theoretical Monte Carlo Markov chain procedure in the framework of graphs. It specifically deals with the construction of a Markov chain whose empirical distribution converges to a given reference one. The Markov chain is constrained over an underlying graph so that states are viewed as vertices, and the transition between two states can have positive probability only in the presence of an edge connecting them. The analysis focuses on the relevant case of support of the target distribution not connected in the graph. Some general arguments on the speed of convergence are also carried out.