{"ID":2850812,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.21950","arxiv_id":"2510.21950","title":"Heaven \u0026 Hell II: Scale Laws and Robustness in One-Step Heaven-Hell Consensus","abstract":"We study Heaven-Hell dynamics, a model for network consensus. A known result establishes an exact one-step convergence threshold for systems with a single uniform hub: the per-node inbound hub weight W suffices if and only if W \u003e= maxrest, the maximum non-hub inbound mass. We develop scale laws and operational refinements that make this threshold robust to tie-breaking policies, node-specific tolerances, targeted seeding, multiple hubs, and asynchronous updates. Our contributions include a conservation-law perspective, parameterized tie policies, tighter pointwise bounds improving on classical worst-case guarantees, one-pass fairness for asynchronous updates, and sufficient conditions for seeded convergence. All proofs are mechanized in Coq, with experiments on rings, grids, scale-free graphs, and heterogeneous weighted graphs validating tightness and gap closures","short_abstract":"We study Heaven-Hell dynamics, a model for network consensus. A known result establishes an exact one-step convergence threshold for systems with a single uniform hub: the per-node inbound hub weight W suffices if and only if W \u003e= maxrest, the maximum non-hub inbound mass. We develop scale laws and operational refineme...","url_abs":"https://arxiv.org/abs/2510.21950","url_pdf":"https://arxiv.org/pdf/2510.21950v1","authors":"[\"Nnamdi Daniel Aghanya\",\"Romain Leemans\"]","published":"2025-10-24T18:24:47Z","proceeding":"cs.SI","tasks":"[\"cs.SI\",\"cs.DC\"]","methods":"[]","has_code":false}