{"ID":2859940,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.04924","arxiv_id":"2510.04924","title":"Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation in Networked Systems","abstract":"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-root 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 non-asymptotic, easy to compute, and certifies the maximum steady-state deviation.","short_abstract":"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, meas...","url_abs":"https://arxiv.org/abs/2510.04924","url_pdf":"https://arxiv.org/pdf/2510.04924v2","authors":"[\"Ardavan Rahimian\"]","published":"2025-10-06T15:35:18Z","proceeding":"eess.SP","tasks":"[\"eess.SP\",\"eess.SY\"]","methods":"[\"Diffusion Model\"]","has_code":false}