Subdiffusive Transport on Graphs

• Subdiffusive transport on graphs is a phenomenon characterized by anomalously slow spreading, where mean-square displacement grows sublinearly (⟨r²(t)⟩ ∝ t^β with 0 < β < 1).
• The framework employs fractional time derivatives, spectral analysis, and topological mechanisms to model dynamics across fermionic, stochastic, and percolation networks.
• Applications span quantum wires, disordered networks, and biological systems, with models unveiling scaling laws in conductance and chemical distances.

Subdiffusive transport on graphs refers to anomalously slow spreading phenomena, characterized by a mean-square displacement (MSD) that grows sublinearly in time, $\langle r^2(t) \rangle \propto t^\beta$ with $0 < \beta < 1$, or by nontrivial scaling of transport observables such as conductance or current with system size. Mechanisms underlying subdiffusive behavior range from memory effects and random temporal inhomogeneity to topological constraints, disorder-induced trapping, and power-law waiting-time distributions. This article surveys the mathematical and physical theory of subdiffusive transport across fermionic, classical, and stochastic network models, as well as its rigorous concentration properties in random graphs, with particular emphasis on recent advances in fractional graph dynamics, many-body systems with ergodic/frozen domains, and percolation frameworks.

1. Mathematical Models of Subdiffusive Transport on Graphs

Subdiffusion is most prominently realized via fractional dynamics on graphs, first and foremost through the time-fractional diffusion equation. On a finite, undirected graph $G=(V,E)$ with combinatorial Laplacian $L=D-A$, Caputo time-fractional diffusion governs the probability vector $u(t)$ as

$$D_t^\alpha u(t) + \theta L u(t) = 0, \quad 0<\alpha<1,$$

where $D_t^\alpha$ is the Caputo fractional derivative,

$$D_t^\alpha f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-\tau)^{-\alpha} f'(\tau)\, d\tau,$$

and $\theta$ is the diffusion rate. The generalized framework incorporates both time- and space-fractionality via the equation

$D_t^\beta p(t) = -K L_d^{(\alpha)} p(t),$

where $L_d^{(\alpha)}$ denotes the transformed $d$-path Laplacian (a weighted sum of $k$-step Laplacians) and $K>0$ is a constant. The propagator is explicitly given by the matrix Mittag-Leffler function $E_\beta(-K L_d^{(\alpha)} t^\beta)$ acting on the initial state (Diaz-Diaz et al., 2022, Deniskin et al., 21 Jan 2026).

In fermionic wire models, subdiffusion emerges from tight-binding Hamiltonians on rectangular lattices. Consider a Hamiltonian $H_W$ built from intra- and inter-chain hopping matrices $H_0$ and $H_1$, with the transport features encoded in the spectrum of $A(\omega) = (-\omega + H_0) H_1^{-1}$, where $\omega$ is the Fermi level (Bhat, 2023).

Stochastic network models, such as facilitated classical hopping subject to disorder and dephasing, or percolation-induced random graphs, yield kinetic equations and master equations for particle occupation that give rise to subdiffusive regimes through power-law waiting times, rare region effects, or constrained graph geometry (Klocke et al., 2021, Can et al., 2024).

2. Spectral Criteria, Memory, and Subdiffusive Exponents

Fractional diffusion on graphs fundamentally encodes memory effects. The subordination principle relates time-fractional diffusion to a random time change of standard Markovian diffusion: a process with generator $-L$ subordinated by an $\alpha$-stable process $S_t$ leads to non-Markovian dynamics with long-tailed waiting times and power-law relaxation,

$$E_\alpha(-t^\alpha L) = \int_0^\infty e^{-sL} g_\alpha(s, t) ds,$$

with $g_\alpha(s, t)$ a probability density tied to the inverse subordinator (Deniskin et al., 21 Jan 2026). As a consequence, algebraic decay of modes,

$$E_\alpha(-\lambda t^\alpha) \sim \frac{1}{\lambda \Gamma(1-\alpha)} t^{-\alpha}$$

dominates at late times, in contrast to the exponential decay in normal diffusion.

The spectral structure of operators determines observable exponents:

• For time-fractional diffusion, $\langle r^2(t) \rangle \sim t^\alpha$, subdiffusive for $\alpha < 1$.
• In generalized space-time fractional models, $\langle r^2(t) \rangle \sim t^\gamma$ with $\gamma = 2\beta/\mu$, where $\mu$ is an effective spatial dispersion exponent; subdiffusion for $2\beta/\mu < 1$ (Diaz-Diaz et al., 2022).

In fermionic wires, the scaling of the two-terminal conductance $T(\omega)$ with longitudinal wire length $N_x$ falls into three regimes, governed by the eigenvalues $\{\lambda_k\}$ of $A(\omega)$:

• Ballistic: $$\exists\,k:|\lambda_k(\omega)|<2 \implies T(\omega) = O(1)$$,
• Subdiffusive: $$\exists\,k^*:|\lambda_{k^*}|=2,\,\forall k\neq k^*:|\lambda_k|>2 \implies T \sim 1/N_x^2$$,
• Exponential: $$\forall k:|\lambda_k|>2\implies T \sim D(\omega) e^{-\alpha N_x}$$.

At the topological transition, where multiple eigenvalues hit $|\lambda|=2$, higher-order subdiffusive scaling ($T \sim 1/N_x^3$) appears (Bhat, 2023).

3. Physical Mechanisms: Trapping, Topology, and Rare-Region Effects

Subdiffusive transport on graphs can arise from multiple physical mechanisms:

• Memory and vertex inhomogeneity: In fractional graph dynamics, heavy-tailed waiting times yield “aging,” with survival at a node $i$ decaying as $S_i(t)\sim (1/d_i\Gamma(1-\alpha)) t^{-\alpha}$, showing pronounced dependence on vertex degree; low-degree nodes act as traps with long dwell times (Deniskin et al., 21 Jan 2026).
• Topological mechanisms: In two-dimensional fermionic wires, subdiffusive channels can be protected by the spectral topology of $A(\omega)$; edge states at $\lambda=2$ yield robust $T \sim 1/N_x^2$ subdiffusion, insensitive to weak disorder as a result of topological protection (Bhat, 2023).
• Disorder and rare thermal regions: In facilitated network models (analogues of many-body localization with dephasing), thermal “bubbles” embedded in frozen domains give rise to a power-law distribution of trapping times, leading to subdiffusive MSD scaling $\langle x^2(t)\rangle \sim t^\mu$, with $0<\mu<1$. The steady-state current scales as $j(L)\sim L^{-\alpha}$ with $\alpha = 1/\mu > 1$ (Klocke et al., 2021).
• Percolation geometry: In supercritical percolation, subdiffusive fluctuations of graph distance are governed by detour radii and the probabilistic structure of open paths; this yields strong concentration of chemical distance and subdiffusive variance scaling of order $n/\log n$ (Can et al., 2024).

4. Observable Manifestations and Scaling Laws

The manifestation of subdiffusive transport is observable in several key quantities:

• Mean-square displacement: For a random walker beginning at node $j_0$, the MSD in time-fractional models satisfies $\langle r^2(t) \rangle \sim t^\beta$, with $\beta<1$ for subdiffusion.
• Conductance and current exponents:
  • In fermionic wires, $T \sim 1/N_x^2$ for protected subdiffusive modes and $T \sim 1/N_x^3$ at topological transitions (Bhat, 2023).
  • In stochastic networks, the stationary current subject to a chemical potential difference across a chain of size $L$ decays as $j(L) \sim L^{-\alpha}$ with $\alpha>1$, reflecting subdiffusion (Klocke et al., 2021).
• Statistical fluctuations: In percolation, the chemical distance between distant points has variance scaling as $O(n/\log n)$, improving upon the classical $O(n)$ “diffusive” bound, indicating “wandering” of geodesics is sublinear (Can et al., 2024).

The following table summarizes governing equations and typical exponents:

Model Class	Governing Equation	Key Subdiffusive Scaling
Fractional graph dynamics	$D_t^\alpha u(t) + L u(t) = 0$	$\langle r^2(t)\rangle \sim t^\alpha$
Fermionic wires	Conductance via NEGF; matrix $A(\omega)$	$T \sim 1/N_x^2$, $1/N_x^3$
Facilitated classical MBL	Master equation for $P({n},t)$	$j(L)\sim L^{-\alpha}$ ($\alpha>1$)
Percolation	Geodesic length, effective radius $R_e$	$\operatorname{Var} D \sim n/\log n$

5. Geometric and Topological Features of Subdiffusive Dynamics

Fractional diffusion naturally defines a time-dependent subdiffusive geometry on the underlying graph. The subdiffusive communicability distance

$$\mathscr{D}_{\alpha, t}(v, w) = E_\alpha(-t^{\alpha} L)_{vv} + E_\alpha(-t^{\alpha} L)_{ww} - 2E_\alpha(-t^{\alpha} L)_{vw}$$

is positive-definite and encodes an evolving metric with shortest paths asymptotically coinciding with the ordinary graph geodesics as $t\to 0$. However, in selecting among multiple geodesics, the subdiffusive metric preferentially weights routes traversing high-degree vertices, indicating a reinforced memory effect and local preference for “hubs” (Deniskin et al., 21 Jan 2026).

In fermionic wires, nontrivial topology of the spectral parameter space (i.e., symmetry-protected zero-modes of $A(\omega)$) underpins robust subdiffusive channels that survive under disorder, mimicking the persistence of topologically protected edge states in topological phases (Bhat, 2023).

In percolation, the renormalization via effective detour radius $R_e$ produces chemical balls with strongly concentrated boundaries, yielding a metric space with subdiffusive concentration properties (Can et al., 2024).

6. Connections, Singular Limits, and Open Problems

Fractional subdiffusion emerges as a singular limit of multi-rate or “multiplicative” diffusion. Any sum-of-exponentials (SOE) solution for conventional diffusion rates can approximate fractional dynamics up to geometric accuracy; in the limit of infinitely many rates, one recovers the Caputo time-fractional evolution with a scale-free, long-memory kernel (Deniskin et al., 21 Jan 2026).

Open questions remain regarding optimal fluctuation exponents in percolation, sensitivity and chaos in chemical distances, and extensions to weighted random graphs and non-Euclidean geometries. Physical realizations span quantum wires, disordered networks, biological transport systems (e.g., protein-DNA sliding with alternating subdiffusive and superdiffusive phases), and models of many-body localization transition (Bhat, 2023, Diaz-Diaz et al., 2022, Klocke et al., 2021).

Future developments will likely refine our understanding of the interplay between memory, topology, disorder, and geometry in driving universal subdiffusive transport across complex networked systems.