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Modern neuroimaging technologies set the stage for studying structural connectivity (SC) and functional connectivity (FC) \textit{in-vivo}. Due to distinct biological wiring underpinnings in SC and FC, however, it is challenging to understand their coupling mechanism using statistical association approaches. We seek to answer this challenging neuroscience question through the lens of a novel perspective rooted in network topology. Specifically, our assumption is that each FC instance is either locally supported by the direct link of SC or collaboratively sustained by a group of alternative SC pathways which form a topological notion of \textit{detour}. In this regard, we propose a new connectomic representation, coined detour connectivity (DC), to characterize the complex relationship between SC and FC by presenting direct FC with the weighted connectivity strength along in-directed SC routes. Furthermore, we present SC-FC Detour Network (SFDN), a graph neural network that integrates DC embedding through a self-attention mechanism, to optimize detour to the extent that the coupling of SC and FC is closely aligned with the evolution of cognitive states. We have applied the concept of DC in network community detection while the clinical value of our SFDN is evaluated in cognitive task recognition and early diagnosis of Alzheimer’s disease. After benchmarking on three public datasets under various brain parcellations, our detour-based computational approach shows significant improvement over current state-of-the-art counterpart methods.