Algebraic Connectivity Reveals Modulated High-Order Functional Networks in Alzheimer's Disease.

Functional MRI is a neuroimaging technique that analyzes the functional activity of the brain by measuring blood-oxygen-level-dependent signals throughout the brain. The derived functional features can be used for investigating brain alterations in neurological and psychiatric disorders. In this work, we employed a hypergraph to model high-order functional relations across brain regions, introducing algebraic connectivity ($a(\mathcal {G})$) for estimating the hyperedge weights. The hypergraph structure was derived from healthy controls to build a common topology across individuals. The considered cohort for subsequent analyses included subjects covering the Alzheimer's disease (AD) continuum, encompassing both mild cognitive impairment and AD patients. Statistical analysis and three classification tasks: HC vs AD, MCI vs AD, and HC vs MCI, were performed to assess differences across the three groups and the potential of the hyperedge weights as functional features. Furthermore, a mediation analysis was performed to evaluate the reliability of the $a(\mathcal {G})$ values, representing functional information as the mediator between tau-PET levels, a key biomarker of AD, and cognitive scores. The proposed approach identified a larger number of hyperedges statistically different across groups compared to state-of-the-art methods. The $a(\mathcal {G})$ hyperedge weights also achieved higher performance in two classification tasks, while similar performance in the third. Finally, two hyperedges belonging to salience/ventral attention and somatomotor networks showed a partial mediation effect between the tau biomarker and cognitive decline. These results suggested that $a(\mathcal {G})$ can be an effective approach for extracting the hyperedge weights, including important functional information that resides in the brain areas forming the hyperedges.