  We study random walks on contingency tables with fixed marginals,
corresponding to a (log-linear) hierarchical model. If the set of allowed moves
is not a Markov basis, then there exist tables with the same marginals that are
not connected. We study linear conditions on the values of the marginals that
ensure that all tables in a given fiber are connected. We show that many
graphical models have the positive margins property, which says that all fibers
with strictly positive marginals are connected by the quadratic moves that
correspond to conditional independence statements. The property persists under
natural operations such as gluing along cliques, but we also construct examples
of graphical models not enjoying this property. We also provide a negative
answer to a question of Engstr\"om, Kahle, and Sullivant by demonstrating that
the global Markov ideal of the complete bipartite graph K_(3,3) is not radical.
  Our analysis of the positive margins property depends on computing the
primary decomposition of the associated conditional independence ideal. The
main technical results of the paper are primary decompositions of the
conditional independence ideals of graphical models of the $N$-cycle and the
complete bipartite graph $K_(2,N-2)$, with various restrictions on the size of
the nodes.
