Powerful digraphs

We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type $p$ possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type $p$ and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (>1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (>1) countable models.