A Wasserstein Weisfeiler-Lehman Kernel Framework for Multimodal Brain Connectome Analysis

Abstract We introduce a Wasserstein Weisfeiler-Lehman (WWL) graph-kernel framework for the joint analysis of weighted structural graphs and continuous node attributes, together with a regularized fractional-Laplacian extension of the underlying graph propagation operator. The proposed fractional operator provides non-local propagation across the entire graph, with influence decaying as a power law with graph distance, while remaining anchored to the subject-specific structural topology. This extends the standard finite-range Weisfeiler Lehman propagation scheme to a controlled long-range setting without introducing an externally prescribed coupling between structural and functional information. We formulate the framework for multimodal brain connectivity by representing each subject as a single attributed graph: diffusion tractography defines the weighted structural edges, while resting-state functional connectivity profiles are encoded as continuous node attributes. We evaluate the method on a synthetic cohort of $900$ subjects in three ordered clinical groups along the Alzheimer's disease spectrum, generated from multiple structural templates and characterised by a spatially structured loss of resting-state network segregation with age. Under nested cross-validation, the WWL kernel achieves a balanced accuracy of $0.651\pm0.028$, significantly outperforming both a raw-feature SVM and a histogram-based WL-kernel baseline. Controlled ablations separate the contribution of the Wasserstein representation from that of structural propagation. Replacing the propagation operator by the identity yields essentially identical classification performance, whereas a structure-only representation approaches chance, showing that the discriminative gain in this benchmark is primarily associated with the Wasserstein point-cloud representation rather than structural propagation. Conversely, experiments across distinct signal regimes show that the fractional extension is signal-dependent: non-local propagation can improve continuous prediction for diffuse whole-brain coupling signals, but provides no consistent advantage for network-segregation signals. We further identify a connectivity artefact induced by fixed-percentile structural sparsification and establish a threshold preserving graph connectivity across subjects. The results clarify the respective roles of Wasserstein geometry, structural propagation, and non-locality in multimodal graph representations and provide a controlled framework for assessing when fractional graph propagation is expected to be beneficial.