Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation in Networked Systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918274686517248 |
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| author | Rahimian, Ardavan |
| author_facet | Rahimian, Ardavan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation in Networked Systems Rahimian, Ardavan Signal Processing Systems and Control 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. |
| title | Steady-State Spread Bounds for Graph Diffusion via Laplacian Regularisation in Networked Systems |
| topic | Signal Processing Systems and Control |
| url | https://arxiv.org/abs/2510.04924 |