First Betti number of the path homology of random directed graphs

TL;DR

This paper investigates the behavior of the first Betti number in the path homology of Erdős-Rényi directed random graphs, revealing phase transitions and providing probabilistic bounds for its positivity across different graph densities.

Contribution

It introduces a new combinatorial condition for analyzing the first Betti number in directed graphs and combines existing methods to characterize its behavior in sparse and dense regimes.

Findings

First Betti number exhibits two phase transitions in Erdős-Rényi directed graphs.

Derived probabilistic bounds for the first Betti number in different density regimes.

Applied techniques to directed flag complex for comparative analysis.

Abstract

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.