Central limit theorem for the first path Betti number
Let $G_n$ be the random digraph on $[n]$ in which each ordered pair $(i, j)$, $i \ne j$, is an edge independently with probability $p_n$; reciprocal edges are allowed. Assume $$n p_n \longrightarrow \infty, \qquad n^{2/3} p_n \longrightarrow 0.$$ Over $\mathbb{Q}$, let $\Lambda_k$ be the vector space with basis all vertex sequences $[v_0, \dots, v_k]$, repetitions allowed, with the boundary $$\partial [v_0, \dots, v_k] = \sum_{i=0}^{k} (-1)^i [v_0, \dots, \widehat{v_i}, \dots, v_k],$$ in which no term with repeated consecutive vertices is discarded (the non-regular convention). Let $A_k \subseteq \Lambda_k$ be spanned by the sequences whose successive ordered pairs are edges of $G_n$, put $\Omega_0 = A_0$ and $\Omega_k = A_k \cap \partial^{-1}(A_{k-1})$ for $k \ge 1$, and let $$B_n = \dim_{\mathbb{Q}} \frac{\ker(\partial : \Omega_1 \to \Omega_0)}{\partial \Omega_2},$$ the rank of the integral non-regular first path homology of $G_n$. Chaplin proved that in this range $\mathbb{E} B_n \sim n(n-1)p_n$ and that $B_n > 0$ with probability tending to one.
Problem. Does every sequence $(p_n)$ satisfying the two conditions obey $$\frac{B_n - \mathbb{E} B_n}{\sqrt{\operatorname{Var}(B_n)}} \xrightarrow{\ d\ } \mathcal{N}(0, 1)?$$
Source: Chaplin (2024), Conjecture B.2. Stated by AIM — Open Applied Problems for the rational non-regular first path Betti number.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AT, CO
- Keywords
- AIM Open Applied Problems, central limit theorems, path homology, random directed graphs
References
@article{chaplin2024,
author = {Chaplin, Thomas},
title = {First {B}etti number of the path homology of random directed graphs},
journal = {J. Appl. Comput. Topol.},
volume = {8},
number = {6},
year = {2024},
pages = {1503--1549},
doi = {10.1007/s41468-022-00108-3},
eprint = {2111.13493},
archivePrefix = {arXiv},
}
@misc{aim-random-digraph-path-betti-central-limit,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Central limit theorem for the first path Betti number}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/650-random-digraph-path-betti-central-limit.md},
note = {Entry 650 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove a central limit theorem for the first non-regular path Betti number of a random digraph with independent ordered edges of probability $p_n$, in the full range $np_n\to\infty$ and $n^{2/3}p_n\to0$. This establishes Chaplin's Conjecture B.2. The variance is asymptotic to $n(n-1)p_n(1-p_n)$. The proof compares centered Betti numbers with the edge count: a monotone family of short two-chain generators controls …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, 650-random-digraph-path-betti-central-limit/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00827, "Central limit theorem for the first path Betti number", clopen.solutions.