Multi-Scale Profiling of Brain Multigraphs by Eigen-based Cross-Diffusion and Heat Tracing for Brain State Profiling

The individual brain can be viewed as a highly-complex multigraph (i.e. a set of graphs also called connectomes), where each graph represents a unique connectional view of pairwise brain region (node) relationships such as function or morphology. Due to its multifold complexity, understanding how brain disorders alter not only a single view of the brain graph, but its multigraph representation at the individual and population scales, remains one of the most challenging obstacles to profiling brain connectivity for ultimately disentangling a wide spectrum of brain states (e.g., healthy vs. disordered). In this work, while cross-pollinating the fields of spectral graph theory and diffusion models, we unprecedentedly propose an eigen-based cross-diffusion strategy for multigraph brain integration, comparison, and profiling. Specifically, we first devise a brain multigraph fusion model guided by eigenvector centrality to rely on most central nodes in the cross-diffusion process. Next, since the graph spectrum encodes its shape (or geometry) as if one can hear the shape of the graph, for the first time, we profile the fused multigraphs at several diffusion timescales by extracting the compact heat-trace signatures of their corresponding Laplacian matrices. Here, we reveal for the first time autistic and healthy profiles of morphological brain multigraphs, derived from T1-w magnetic resonance imaging (MRI), and demonstrate their discriminability in boosting the classification of unseen samples in comparison with state-of-the-art methods. This study presents the first step towards hearing the shape of the brain multigraph that can be leveraged for profiling and disentangling comorbid neurological disorders, thereby advancing precision medicine.