  Mostly acyclic directed networks, treated mathematically as directed graphs,
arise in machine learning, biology, social science, physics, and other
applications. Newman [1] has noted the mathematical challenges of such
networks. In this series of papers, we study their connectivity properties,
focusing on three types of phase transitions that affect horizon sizes for
typical nodes. The first two types involve the familiar emergence of giant
components as average local connectivity increases, while the third type
involves small-world horizon growth at variable distance from a typical node.
In this first paper, we focus on qualitative behavior, simulations, and
applications, leaving formal considerations for subsequent papers. We explain
how such phase transitions distinguish deep neural networks from shallow
machine learning architectures, and propose hybrid local/random network designs
with surprising connectivity advantages. We also propose a small-world approach
to the horizon problem in the cosmology of the early universe as a novel
alternative to the inflationary hypothesis of Guth and Linde.
