Topologically Invariant Permutation Test

TL;DR

This paper presents a novel permutation test leveraging topological features and Wasserstein distances to compare brain networks, demonstrating improved accuracy and robustness in detecting topological differences in fMRI data.

Contribution

The paper introduces a topologically invariant permutation test using persistent diagrams and heat kernel expansion, providing theoretical guarantees and superior performance over existing methods.

Findings

Effective detection of topological differences in simulated networks.

Superior performance compared to traditional tests.

Identified significant topological differences in ADHD fMRI data.

Abstract

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.