  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.
