Instance-Adaptive Deviation Bounds for Graph-Diffused Allocation in Networked Systems

Graph-structured designs often require a seeded nonnegative profile to remain well controlled under diffusion, so that spreading across the graph is limited while smoothness or regularity is preserved. This raises a practical question: how large must the Laplacian regularisation weight be to certify that the eventual diffused steady state stays within a prescribed deviation from the original design? For row-stochastic and generally nonsymmetric diffusion operators, standard worst-case certificates are often conservative as they rely on global mismatch and dimensional surrogates rather than on the realised profile itself. The work derives a proof-consistent instance-adaptive certificate that combines a symmetric certificate operator, a non-symmetry remainder, a profile-dependent diagonal-mismatch term and an effective support size. The resulting expression remains closed form, supports fixed-instance inversion and one-shot pilot-based tuning, and is compatible with practically relevant nonsymmetric diffusion models. The affine diffusion model is interpreted as a graph-side mixing/restart mechanism rather than as a mass-conserving physical transport law; thus, norm stability and conservation are kept conceptually separate. The proposed framework can further be applied to a number of applications, including graph-based beamforming, wireless resource allocation and other networked diffusion settings where certified control of spread is required without repeated parameter sweeps.