  We analyze the properties of degree-preserving Markov chains based on
elementary edge switchings in undirected and directed graphs. We give exact yet
simple formulas for the mobility of a graph (the number of possible moves) in
terms of its adjacency matrix. This formula allows us to define acceptance
probabilities for edge switchings, such that the Markov chains become
controlled Glauber-type detailed balance processes, designed to evolve to any
required invariant measure (representing the asymptotic frequencies with which
the allowed graphs are visited during the process).

Disordered Systems and Neural Networks