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.

翻译：功能脑网络展现出反映神经组织结构的拓扑特征，然而对这些网络进行统计比较面临多重挑战。本文提出一种用于检测拓扑不等价的拓扑不变置换检验方法。在拓扑等价条件下，拓扑特征可在组间独立置换而不破坏个体网络结构。检验统计量基于持续同调图的$2$-Wasserstein距离，采用闭式解计算。为在保持拓扑结构的同时降低脑连接性变异，本文应用霍奇拉普拉斯算子的热核展开方法，其中带宽参数$t$控制扩散强度。理论结果通过最优希尔伯特空间投影保证了方差缩减。在不同网络拓扑结构上的仿真实验表明，相较于传统双样本检验与替代性度量方法，本方法具有更优性能。应用于多模态ADHD治疗研究的静息态fMRI数据时，该方法成功检测出大麻使用者与非使用者间显著的拓扑差异。