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Graph theoretical analyses have become standard tools in modeling functional and anatomical connectivity in the brain. With the advent of connectomics, the primary graphs or networks of interest are structural connectome (derived from DTI tractography) and functional connectome (derived from resting-state fMRI). However, most published connectome studies have focused on either structural or functional connectome, yet complementary information between them, when available in the same dataset, can be jointly leveraged to improve our understanding of the brain. To this end, we propose a function-constrained structural graph variational autoencoder (FCS-GVAE) capable of incorporating information from both functional and structural connectome in an unsupervised fashion. This leads to a joint low-dimensional embedding that establishes a unified spatial coordinate system for comparing across different subjects. We evaluate our approach using the publicly available OASIS-3 Alzheimer’s disease (AD) dataset and show that a variational formulation is necessary to optimally encode functional brain dynamics. Further, the proposed joint embedding approach can more accurately distinguish different patient sub-populations than approaches that do not use complementary connectome information.