Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation

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.