  Path homology is a topological invariant for directed graphs, which is
sensitive to their asymmetry and can discern between digraphs which are
indistinguishable to the directed flag complex. In Erd\"os-R\'enyi directed
random graphs, the first Betti number undergoes two distinct transitions,
appearing at a low-density boundary and vanishing again at a high-density
boundary. Through a novel, combinatorial condition for digraphs we describe
both sparse and dense regimes under which the first Betti number of path
homology is zero with high probability. We combine results of Grigor'yan et
al., regarding generators for chain groups, with methods of Kahle and Meckes in
order to determine regimes under which the first Betti number is positive with
high probability. Together, these results describe the gradient of the lower
boundary and yield bounds for the gradient of the upper boundary. With a view
towards hypothesis testing, we obtain tighter bounds on the probability of
observing a positive first Betti number in a high-density digraph of finite
size. For comparison, we apply these techniques to the directed flag complex
and derive analogous results.

Algebraic Topology;Combinatorics;Probability