Directed graphs are ubiquitous models for networks, and topological spaces they generate, such as the directed flag complex, have become useful objects in applied topology. Here the directed cliques form directed simplices. We extend Atkin's theory of q-connectivity to the case of directed simplices. This results in a preorder where simplices are related by sequences of simplices that share a q-face with respect to directions specified by chosen face maps. We leverage the Alexandroff equivalence between preorders and topological spaces to introduce a new class of topological spaces for directed graphs, enabling us to assign new homotopy types different from those of directed flag complexes as seen by simplicial homology. We further introduce simplicial path analysis enabled by the connectivity preorders. As an application we characterize structural differences between various brain networks with respect to their simplicial paths.