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arXiv · 2510.21950

Heaven & Hell II: Scale Laws and Robustness in One-Step Heaven-Hell Consensus

Nnamdi Daniel Aghanya·Romain Leemans
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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 >= 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

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Nnamdi Daniel Aghanya, Romain Leemans. 2025-10-24. Heaven & Hell II: Scale Laws and Robustness in One-Step Heaven-Hell Consensus. https://arxiv.org/abs/2510.21950

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