  This study, we introduce a novel Topological Cycle Graph Attention Network
(CycGAT), designed to delineate a functional backbone within brain functional
graph--key pathways essential for signal transmissio--from non-essential,
redundant connections that form cycles around this core structure. We first
introduce a cycle incidence matrix that establishes an independent cycle basis
within a graph, mapping its relationship with edges. We propose a cycle graph
convolution that leverages a cycle adjacency matrix, derived from the cycle
incidence matrix, to specifically filter edge signals in a domain of cycles.
Additionally, we strengthen the representation power of the cycle graph
convolution by adding an attention mechanism, which is further augmented by the
introduction of edge positional encodings in cycles, to enhance the topological
awareness of CycGAT. We demonstrate CycGAT's localization through simulation
and its efficacy on an ABCD study's fMRI data (n=8765), comparing it with
baseline models. CycGAT outperforms these models, identifying a functional
backbone with significantly fewer cycles, crucial for understanding neural
circuits related to general intelligence. Our code will be released once
accepted.
