  Heteroclinic connections are trajectories that link invariant sets for an
autonomous dynamical flow: these connections can robustly form networks between
equilibria, for systems with flow-invariant spaces. In this paper we examine
the relation between the heteroclinic network as a flow-invariant set and
directed graphs of possible connections between nodes. We consider realizations
of a large class of transitive digraphs as robust heteroclinic networks and
show that although robust realizations are typically not complete (i.e. not all
unstable manifolds of nodes are part of the network), they can be almost
complete (i.e. complete up to a set of zero measure within the unstable
manifold) and equable (i.e. all sets of connections from a node have the same
dimension). We show there are almost complete and equable realizations that can
be closed by adding a number of extra nodes and connections. We discuss some
examples and describe a sense in which an equable almost complete network
embedding is an optimal description of stochastically perturbed motion on the
network.
