Heaven & 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 >= 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
Cite
@article{arxiv.2510.21950,
title = {Heaven & Hell II: Scale Laws and Robustness in One-Step Heaven-Hell Consensus},
author = {Nnamdi Daniel Aghanya and Romain Leemans},
journal= {arXiv preprint arXiv:2510.21950},
year = {2025}
}
Comments
12 pages, 1 figure