Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups and 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. We prove that Mittag-Leffler graph dynamics admit an exact convex, mass-preserving representation as a superposition of classical heat semigroups evaluated at rescaled times, revealing fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales. 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 particles to locally discover global shortest paths while favoring high-degree regions. Finally, we show that time-fractional diffusion arises as a singular limit of multi-rate diffusion.

💡 Research Summary

This paper investigates subdiffusive transport on graphs through the lens of time‑fractional diffusion equations, revealing that the phenomenon is fundamentally a memory‑driven process rather than a simple slowdown of classical diffusion. By replacing the ordinary time derivative in the graph diffusion equation with a Caputo fractional derivative of order (0<\alpha<1), the authors obtain the governing equation
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