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.

• Full paper

Related research projects

SNANDOG

Spatiotemporal Nonlocal and Non-conservative Diffusion on Graphs

P.I.: Ernesto Estrada

Complex systems permeate biological life, society, and modern infrastructures. Their functioning depends primarily on the dynamic processes that take place between interacting entities that form network structures. Such structures can be appropriately …