  Graphical models are popular statistical tools which are used to represent
dependent or causal complex systems. Statistically equivalent causal or
directed graphical models are said to belong to a Markov equivalent class. It
is of great interest to describe and understand the space of such classes.
However, with currently known algorithms, sampling over such classes is only
feasible for graphs with fewer than approximately 20 vertices. In this paper,
we design reversible irreducible Markov chains on the space of Markov
equivalent classes by proposing a perfect set of operators that determine the
transitions of the Markov chain. The stationary distribution of a proposed
Markov chain has a closed form and can be computed easily. Specifically, we
construct a concrete perfect set of operators on sparse Markov equivalence
classes by introducing appropriate conditions on each possible operator.
Algorithms and their accelerated versions are provided to efficiently generate
Markov chains and to explore properties of Markov equivalence classes of sparse
directed acyclic graphs (DAGs) with thousands of vertices. We find
experimentally that in most Markov equivalence classes of sparse DAGs, (1) most
edges are directed, (2) most undirected subgraphs are small and (3) the number
of these undirected subgraphs grows approximately linearly with the number of
vertices. The article contains supplement arXiv:1303.0632,
http://dx.doi.org/10.1214/13-AOS1125SUPP
