Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation in Networked Systems

We study how far a diffusion process on a graph can deviate from a designed starting pattern when the pattern is generated via 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 (with an inverse square law). This leads to a 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 nonasymptotic, easy to compute and certifies the maximum steady‐state deviation.