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

TL;DR

This paper extends the understanding of Heaven-Hell consensus dynamics by developing robust scale laws and refinements that accommodate various practical complexities, with formal proofs and experimental validation.

Contribution

It introduces a conservation-law perspective and operational refinements that make the convergence threshold robust to multiple real-world factors.

Findings

Robust scale laws for Heaven-Hell consensus thresholds

Tighter bounds and fairness guarantees for asynchronous updates

Experimental validation on diverse graph topologies

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