Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups Incorporating Memory

Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, and breaks Markovianity while preserving linearity and mass conservation. While the subordination representation and complete monotonicity properties of the Mittag-Leffler function are classical, we develop a graph-based synthesis in which Mittag-Leffler dynamics admit an exact convex, mass-preserving representation as a superposition of Laplacian semigroups evaluated at rescaled times. This perspective reveals fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales and enables new structural and dynamical interpretations of graphs. This framework uncovers heterogeneous, vertex-dependent memory effects and induces transport biases absent in classical diffusion, including algebraic relaxation, degree-dependent waiting times, and early-time asymmetries between sources and neighbors. These features define a subdiffusive geometry on graphs, enabling the recovery of global shortest paths, in contrast to the graph exploration of diffusive geometry, while simultaneously favoring high-degree regions. Finally, we show that time-fractional diffusion can be interpreted as a singular limit of multi-rate diffusion, in an appropriate asymptotic sense.