  We introduce a statistical mechanics formalism for the study of constrained
graph evolution as a Markovian stochastic process, in analogy with that
available for spin systems, deriving its basic properties and highlighting the
role of the `mobility' (the number of allowed moves for any given graph). As an
application of the general theory we analyze the properties of
degree-preserving Markov chains based on elementary edge switchings. We give an
exact yet simple formula for the mobility in terms of the graph's adjacency
matrix and its spectrum. 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). As a corollary we also
derive a condition in terms of simple degree statistics, sufficient to
guarantee that, in the limit where the number of nodes diverges, even for
state-independent acceptance probabilities of proposed moves the invariant
measure of the process will be uniform. We test our theory on synthetic graphs
and on realistic larger graphs as studied in cellular biology.
