From nodes to pathways: an edge-centric model of brain function-structure coupling via constrained Laplacians

Understanding how functional relationships relate to the brains structural architecture remains a central challenge in network neuroscience. Many Laplacian-based approaches describe function-structure coupling at the node level, which can make it difficult to identify the specific anatomical pathways that support observed functional relationships. This work introduces a constrained Laplacian formulation that incorporates externally specified pairwise functional relationships and yields a nodal field whose graph-gradient representation produces edge-level quantities describing how the structural network accommodates these relationships. The method is implemented using Modified Nodal Analysis, enabling efficient computation on large connectomes. Given an observed pattern of functional associations and a structural connectivity graph, the proposed framework estimates which structural edges are most consistent with supporting the imposed pattern. The framework is demonstrated in multiple settings, including a single-subject example, a controlled diffusion phantom, an in-silico function-structure simulation, and analyses of Human Connectome Project data at both group level (207 subjects) and test-retest conditions (three subjects). Across these applications, the method produces an edge-level representation of function-structure coupling, enabling pathway-specific analysis of brain connectivity.