Persistent homology (PH) characterizes the shape of brain networks through persistence features. Group comparison of persistence features from brain networks can be challenging as they are inherently heterogeneous. A recent scale-space representation of persistence diagrams (PDs) through heat diffusion reparameterizes them using a finite number of Fourier coefficients with respect to the Laplace--Beltrami (LB) eigenfunction expansion of the domain, providing a powerful vectorized algebraic representation for group comparisons. In this study, we develop a transposition-based permutation test for comparing multiple groups of PDs using heat-diffusion estimates. We evaluate the empirical performance of the spectral transposition test in capturing within- and between-group similarity and dissimilarity under varying levels of topological noise and cycle location variability. In application, we propose a topological lesion symptom mapping (TLSM) method based on the proposed framework. The method is applied to resting-state functional brain networks of individuals with post-stroke aphasia to identify characteristic cycles associated with varying levels of speech-language impairment.

翻译：持久同调通过持久性特征刻画脑网络的拓扑形态。由于脑网络固有的异质性，对源自不同脑网络的持久性特征进行组间比较具有挑战性。近期通过热扩散构建的持久图尺度空间表示方法，借助定义域上的拉普拉斯-贝尔特拉米特征函数展开，使用有限个傅里叶系数对持久图进行重参数化，从而为组间比较提供了强大的向量化代数表示。本研究开发了一种基于转置的置换检验方法，利用热扩散估计对多组持久图进行比较。我们评估了谱转置检验在不同拓扑噪声水平和环位置变异程度下捕捉组内相似性与组间差异性的实证性能。在应用层面，我们基于所提框架提出了一种拓扑病灶-症状映射方法。该方法应用于卒中后失语症患者的静息态功能脑网络，旨在识别与不同言语障碍程度相关的特征性拓扑环结构。