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

Nnamdi Daniel Aghanya, Romain Leemans

TL;DR

The work addresses deterministic consensus under Heaven-Hell dynamics on weighted directed graphs with a single hub, deriving an exact one-step threshold $W \ge \max_{v \neq g} (\mathrm{rest\_weight}(v) - \tau(v))$ and developing scale laws to enhance robustness to tolerances, tie policies, seeded/multi-hub setups, and asynchronous updates. It combines a conservation-law perspective with a majority-form view and monotonicity principles to obtain tight, per-node thresholds and exact seeded/multi-hub criteria, all formalized in Coq and validated on rings, grids, scale-free, and heterogeneous graphs. Key contributions include the Pointwise Deg–Max Bound, exact seeded convergence conditions, one-pass fairness guarantees, and extensive empirical validations demonstrating tightness and practical relevance. The results offer precise, verifiable guarantees for robust, scalable consensus in networked systems and provide a foundation for future exploration of multi-step policies and dynamic networks.

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

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

TL;DR

The work addresses deterministic consensus under Heaven-Hell dynamics on weighted directed graphs with a single hub, deriving an exact one-step threshold and developing scale laws to enhance robustness to tolerances, tie policies, seeded/multi-hub setups, and asynchronous updates. It combines a conservation-law perspective with a majority-form view and monotonicity principles to obtain tight, per-node thresholds and exact seeded/multi-hub criteria, all formalized in Coq and validated on rings, grids, scale-free, and heterogeneous graphs. Key contributions include the Pointwise Deg–Max Bound, exact seeded convergence conditions, one-pass fairness guarantees, and extensive empirical validations demonstrating tightness and practical relevance. The results offer precise, verifiable guarantees for robust, scalable consensus in networked systems and provide a foundation for future exploration of multi-step policies and dynamic networks.

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
Paper Structure (21 sections, 10 theorems, 7 equations, 1 figure)

This paper contains 21 sections, 10 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: The Heaven-Hell Model
  3. Notation.
  4. A Conservation Law and a Majority-Form View
  5. Monotonicity Principles
  6. State Monotonicity.
  7. Tolerance Monotonicity.
  8. Tie Policies and Exact Characterisations
  9. Scale Laws: Exact Thresholds and Tighter Bounds
  10. Pointwise Deg–Max Bound.
  11. Seeding and Multi-Hub Variants
  12. Asynchronous Schedules: One-Pass Fairness
  13. Experiments
  14. Regular Families.
  15. Heterogeneous Weighted Graphs.
  16. ...and 6 more sections

Key Result

Lemma 1

For any state $s$ and node $v$, the scores after forcing the hub are related by:

Figures (1)

  • Figure 1: Dynamics and scale effects. (A) One-step success probability rises sharply at $W/W^*=1$ on rings and grids. (B) On heterogeneous graphs, the classical deg$\cdot$wmax bound diverges with indegree $m$ while the pointwise bound stays tight. (C) Equal multi-hub splits reduce per-hub weight as $1/h$; annotated examples match the text. (D) Under domination, asynchronous one-pass schedules monotonically drive the fraction of Glory to 1.

Theorems & Definitions (19)

  • Lemma 1: Score Conservation
  • proof
  • Theorem 2: Majority Form with Tolerance
  • proof
  • Lemma 3: Tie Policies
  • Theorem 4: Uniform Hub with Tolerance (Exact)
  • proof
  • Corollary 5: Uniform Hub, No Tolerance
  • Theorem 6: Exact One-Step Criterion under Seeded Forcing
  • proof
  • ...and 9 more