# Topologically Invariant Permutation Test

arXiv: https://arxiv.org/abs/2511.06153 · HTML: https://mlsanity.com/paper/2511.06153/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2025.

## Built on by (most cited first, 1 of 1)

- [Cycles Communities from the Perspective of Dendrograms and Gradient Sampling](https://mlsanity.com/paper/2512.12974/) (2025)

## Abstract (arXiv)

Functional brain networks exhibit topological structures that reflect neural organization; however, statistical comparison of these networks is challenging for several reasons. This paper introduces a topologically invariant permutation test for detecting topological inequivalence. Under topological equivalence, topological features can be permuted separately between groups without distorting individual network structures. The test statistic uses 2-Wasserstein distances on persistent diagrams, computed in closed form. To reduce variability in brain connectivities while preserving topology, heat kernel expansion on the Hodge Laplacian is applied with bandwidth t controlling diffusion intensity. Theoretical results guarantee variance reduction through optimal Hilbert space projection. Simulations across diverse network topologies show superior performance compared to conventional two-sample tests and alternative metrics. Applied to resting-state fMRI data from the Multimodal Treatment of ADHD study, the method detects significant topological differences between cannabis users and non-users.