Abstract
Functional brain connectivity analysis has traditionally relied on graph-theoretic models that characterize the brain as a set of nodes and edges. However, conventional network analysis typically requires arbitrary thresholding of connection weights to construct binary graphs, discarding critical information and introducing systemic biases. In this paper, we propose a threshold-free computational framework based on persistent homology—a mathematical tool from topological data analysis (TDA)—to analyze weighted functional brain networks. By tracking the birth and death of topological features across a continuous filtration of simplicial complexes, our method captures multi-dimensional connectivity structures that traditional graph metrics fail to detect. We apply this framework to functional magnetic resonance imaging (fMRI) data from patients with Alzheimer's disease (AD) and healthy controls (HC). Our results demonstrate that persistent homology successfully identifies disrupted global integration and localized cycle alterations in the AD cohort. Specifically, the area under the Betti curves (β0 and β1) serves as a robust topological biomarker, achieving a classification accuracy of 88.5% using a support vector machine, significantly outperforming traditional threshold-dependent graph metrics. This work demonstrates the power of algebraic topology in uncovering complex, multi-scale organizational principles of the human brain.
