  Typed metagraphs are defined as hypergraphs with types assigned to hyperedges
and their targets, and the potential to have targets of hyperedges connect to
whole links as well as targets. Directed typed metagraphs (DTMGs) are
introduced via partitioning the targets of each edge in a typed metagraph into
input, output and lateral sets; one can then look at "metapaths" in which
edges' output-sets are linked to other edges' input-sets. An initial algebra
approach to DTMGs is presented, including introduction of constructors for
building up DTMGs and laws regarding relationships among multiple ways of using
these constructors. A menagerie of useful morphism types is then defined on
DTMGs (catamorphisms, anamorphisms, histomorphisms, futumorphisms,
hylomorphisms, chronomorphisms, metamorphisms and metachronomorphisms),
providing a general abstract framework for formulating a broad variety of
metagraph operations. Deterministic and stochastic processes on typed
metagraphs are represented in terms of forests of DTMGs defined over a common
TMG, where the various morphisms can be straightforwardly extended to these
forests. A variation of the approach to undirected typed metagraphs is
presented; and it is indicated how the framework outlined can applied to
realistic metagraphs involving complexities like dependent and probabilistic
types, multidimensional values and dynamic processing including insertion and
deletion of edges.
