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Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups and Memory

Nikita Deniskin, Ernesto Estrada

TL;DR

The paper develops a rigorous framework for subdiffusion on graphs by viewing fractional diffusion as a memory-laden diffusion with a random clock. It shows that Mittag-Leffler graph dynamics admit a convex, mass-preserving representation as a superposition of classical heat semigroups at rescaled times, enabling efficient sum-of-exponentials (SOE) approximations. The approach reveals vertex-dependent memory, algebraic relaxation, and memory-driven path selection, including subdiffusive distances and shortest paths that increasingly prefer high-degree regions. It also places fractional diffusion within a broader hierarchy of diffusion models via operator-valued memory kernels and multi-rate diffusion limits, linking theory, numerics, and physical interpretation. Together, these results clarify how memory reshapes transport on graphs and provide practical tools for simulating and understanding subdiffusive processes 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.

Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups and Memory

TL;DR

The paper develops a rigorous framework for subdiffusion on graphs by viewing fractional diffusion as a memory-laden diffusion with a random clock. It shows that Mittag-Leffler graph dynamics admit a convex, mass-preserving representation as a superposition of classical heat semigroups at rescaled times, enabling efficient sum-of-exponentials (SOE) approximations. The approach reveals vertex-dependent memory, algebraic relaxation, and memory-driven path selection, including subdiffusive distances and shortest paths that increasingly prefer high-degree regions. It also places fractional diffusion within a broader hierarchy of diffusion models via operator-valued memory kernels and multi-rate diffusion limits, linking theory, numerics, and physical interpretation. Together, these results clarify how memory reshapes transport on graphs and provide practical tools for simulating and understanding subdiffusive processes 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.
Paper Structure (51 sections, 20 theorems, 197 equations, 4 figures, 3 tables)

This paper contains 51 sections, 20 theorems, 197 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Subdiffusion: physical mechanisms and mathematical models
  3. Contributions of the paper
  4. Preliminaries
  5. Notation and assumptions
  6. Subordination identity and the M--Wright density
  7. Complete monotonicity and Bernstein's theorem.
  8. Terminology: subordination and the random clock
  9. The time--changed process $X_{E_{t}}$
  10. Sum of Exponentials approximation
  11. SOE via log--trapezoidal quadrature
  12. SOE discretization via subordination: quadrature weights, mass preservation, and vertex survivals
  13. Tail bounds for the subordination integral and window selection
  14. Left tail (small $\theta$)
  15. Right tail (large $\theta$)
  16. ...and 36 more sections

Key Result

Proposition 2.1

Fix $0<\alpha<1$. The scalar function $\lambda\rightarrow E_{\alpha}(-t^{\alpha}\lambda)$ is completely monotone on $[0,\infty)$. There is a probability density $M_{\alpha}(\theta)$ on $(0,\infty)$ such that and in the operator case (Bochner integral): Furthermore, $E_{\alpha}(-t^{\alpha}L)$ is self-adjoint, positive, and $\|E_{\alpha}(-t^{\alpha}L)\|_{2}=1$. Moreover, if $L\mathbf{1}=0$ then $E

Figures (4)

  • Figure 5.1: Base-10 logarithm of the scalar error of the SOE approximation for different times $t$ and number of addends $J$ for an Erdős--Rényi (ER) random graph with $250$ vertices and $1000$ edges. The plots use $\alpha=0.8$ (A), $\alpha=0.5$ (B), and $\alpha=0.25$ (C).
  • Figure 6.1: (A) Illustration of all topological shortest paths between a given pair of vertices in the example Gabriel graph. Coloring is by edge degrees. Shortest paths of SOE approximation for $J=1,10,20,40$ (B, C, D, E respectively) and $\alpha=0.85$. (F) Subdiffusive shortest path between the same pair of vertices as before obtained from the Mittag-Leffler function. In (B)-(F) the paths are colored according to the time, which is taken as $0.1\leq t\leq1000$ with 300 steps in the interval, at which they are observed and the thickness of the edges in proportional to the number of times they are by the corresponding paths.
  • Figure 6.2: (A) relative (blue) and mass (orange) errors of the probing vector $u(t)$. Comparison of the solutions dynamics with exact Mittag-Leffler and the SOE approximation with $J=5$. (B) average on time of the operator (green), relative (blue), and mass (orange) errors as function of $J$. Average is taken on 300 time instants in the interval $0.1\leq t\leq1000$.
  • Figure 6.3: Comparison of the average Levenshtein distance between the shortest subdiffusive paths and the topological shortest path in the example Gabriel graph, for five values of $\alpha$. The average is taken on 300 time instants in the interval $0.1\leq t\leq1000$.

Theorems & Definitions (51)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 41 more