Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation

TL;DR

This paper addresses how far a diffusion process on a graph can drift from an engineered starting pattern when that pattern is produced via Laplacian regularisation. It derives a non-asymptotic, instance-specific bound on the relative steady-state spread $\xi$, proving a clear $O(\mu^{-1/2})$ decay term in addition to an irreducible degree-driven floor. The bound separates a $d_{\max}$-type baseline from a design-controlled term that scales with the regularisation strength $\mu$, and provides a closed-form corollary to map a target spread to a required $\mu$. The results yield a practical tuning rule for Laplacian-regularised designs in scenarios like array beamforming and extend to any graph-diffusion setting with a Laplacian prior, offering actionable guidance for controlling pattern leakage in networked systems. Overall, the work delivers a robust, computable certificate that links graph topology, diffusion dynamics, and regularisation strength to steady-state performance, with implications for RIS/IRS, sensor networks, and graph neural networks.

Abstract

We study how far a diffusion process on a graph can drift from a designed starting pattern when that pattern is produced using Laplacian regularisation. Under standard stability conditions for undirected, entrywise nonnegative graphs, we give a closed-form, instance-specific upper bound on the steady-state spread, measured as the relative change between the final and initial profiles. The bound separates two effects: (i) an irreducible term determined by the graph's maximum node degree, and (ii) a design-controlled term that shrinks as the regularisation strength increases (following an inverse square-root law). This leads to a simple design rule: given any target limit on spread, one can choose a sufficient regularisation strength in closed form. Although one motivating application is array beamforming, where the initial pattern is the squared magnitude of the beamformer weights, the result applies to any scenario that first enforces Laplacian smoothness and then evolves by linear diffusion on a graph. Overall, the guarantee is non-asymptotic, easy to compute, and certifies how much steady-state deviation can occur.

Figures (3)

• Figure 1: Setup at a glance. A Laplacian-regularised design produces $\mathbf p_0=|\mathbf w^\star|^2$, which then diffuses on the graph to $\mathbf p_\infty$. Theorem \ref{['thm:spreadBound']} controls the spreading $\xi$ via instance constants, with an explicit $1/\sqrt{\mu}$ decay.
• Figure 2: Empirical spreading vs. theoretical bound (log--log). Measured $\xi(\mu)$ (blue) decreases with $\mu$ and remains below the instance-wise upper bound from Theorem \ref{['thm:spreadBound']} (orange dashed). A dash--dot reference line of slope $-1/2$ highlights the predicted $O(\mu^{-1/2})$ scaling. The dotted line marks the feasibility floor $C(\rho,\mathbf G)\,d_{\max}$ for the bound. Over this $\mu$ range, the $\mu^{-1/2}$ term dominates, so the bound is nearly linear on log--log axes; it bends toward $C(\rho,\mathbf G)\,d_{\max}$ only for much larger $\mu$.
• Figure 3: Effect of propagation factor $\rho$ at fixed $\mu=10$. Both the bound and the empirical $\xi$ increase monotonically with $\rho$, consistent with $C(\rho,\mathbf G)=\frac{(1-\rho)\rho}{1-\rho\|\mathbf G\|_2}$. For small $\rho$, the measured $\xi$ is close to zero because diffusion exerts only a weak push on the smoothed initial profile $\mathbf p_0$, while the worst-case bound varies according to $C(\rho,\mathbf G)$.

Theorems & Definitions (8)

• Theorem 1: Interference-spreading bound
• proof
• Remark 1: Euclidean normalisation variant
• Remark 2: Physical interpretation
• Remark 3: Bound tightness and normalisation choice
• Corollary 1: Design guideline
• Remark 4: Design rule under $\|\mathbf w\|_2=1$
• Remark 5: Bend point of the bound