Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups and Memory

2026-01-21

AI Summary

This study addresses the limited understanding of the structural and dynamical mechanisms underlying fractional diffusion on graphs, particularly its memory effects and non-Markovian character. The process is modeled as a memory-driven dynamics governed by a random time change, incorporating heavy-tailed waiting times to preserve linearity and mass conservation. The work establishes its equivalence to a convex combination of multiscale classical heat semigroups. Through an integrated theoretical framework combining Mittag-Leffler graph dynamics, Laplacian semigroup superposition, time rescaling, and multi-rate diffusion limits, it uniquely characterizes vertex-dependent heterogeneous memory effects and early-time propagation asymmetry between source and neighboring nodes. The resulting subdiffusive geometry not only reproduces non-classical transport phenomena—such as algebraic relaxation and degree-dependent waiting times—but also demonstrates that fractional diffusion emerges as a singular limit of multi-rate diffusion, effectively capturing global shortest paths and exhibiting preference for highly connected regions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• fractional diffusion
• subdiffusion
• graph dynamics
• memory effects
• non-Markovian processes

Innovation

• fractional diffusion
• graph dynamics
• memory effects
• Laplacian semigroups
• time-fractional equations