Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups and Memory

TL;DR

This paper reveals that subdiffusive transport on graphs can be understood as a superposition of classical heat semigroups with memory effects, leading to heterogeneous, vertex-dependent dynamics and new insights into fractional diffusion.

Contribution

It provides an exact convex representation of fractional graph dynamics as a superposition of classical diffusions, uncovering the structure of memory effects and transport biases.

Findings

Fractional diffusion can be represented as a superposition of heat semigroups.

Memory effects induce degree-dependent waiting times and asymmetries.

Subdiffusive geometry enables local discovery of global shortest paths.

Abstract

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.