Standard diffusion processes with isotropic noise corruption are not spectrally neutral. They inherit a coarse-to-fine generation hierarchy from the nonuniform data spectrum, because each frequency band loses signal-to-noise ratio (SNR) at its own rate. For signals on a graph, the relevant spectrum is that of the graph shift operator (GSO). A natural construction, therefore, builds both the diffusion dynamics and the denoiser from graph filters of the GSO, and any such design also takes a position on this hierarchy. We expose this hierarchy as a tunable axis through a family of colored score-based diffusions. Fixing the terminal prior to a Gaussian colored by the GSO couples the drift and diffusion through a positive spectral rate, which acts as a clock for each graph-frequency band. The clock sets when a band loses and recovers information, and the forward processes of standard and certain graph-aware diffusions appear as members of this family. A GSO-valued EDM preconditioning splits the denoiser into an analytic Wiener head that carries the Gaussian second-order structure and a graph neural network (GNN) that learns the remaining non-Gaussian dependencies. Numerical experiments show that colored diffusions reach a given generation quality in far fewer sampling steps than standard diffusion.

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