A Density-Delay Law for Stable Event-Driven State Progression in Open Distributed Systems

2026-03-23

AI Summary

This work addresses the challenge of fork formation due to network latency in open distributed systems, which threatens system stability under concurrent proposals. The authors propose a “density–delay law,” asserting that the product of proposal density and propagation delay must remain O(1) to guarantee bounded fork depth. From this principle, they derive a scaling rule: per-node proposal intensity should inversely scale with the number of nodes. They further reveal that Bitcoin’s difficulty adjustment mechanism inherently functions as a regulator of event density. By modeling proposal overlaps via Poisson processes and fork evolution through birth–death processes, the study combines theoretical analysis with large-scale simulations to validate the universality of this scaling law across varying network sizes and latency conditions.

Abstract

Distributed systems in which concurrent proposals are mutually exclusive face a fundamental stability constraint under network delay. In open systems where global state progression is event-driven rather than round-driven, propagation delay creates a conflict window within which overlapping proposals may generate competing branches. This paper derives a density-delay law for such exclusive state progression processes. Under independent proposal arrivals and bounded propagation delay, overlap is approximated by a Poisson model and fork depth is represented by a birth-death process. The analysis shows that maintaining bounded fork depth as the number of participants grows requires the density-delay product $λΔ$ to remain $O(1)$, implying that aggregate proposal intensity must stay bounded and yielding an inverse-scaling law $g(N)=O(1/N)$ at the unit level. Simulation experiments across varying network sizes and propagation delays align with a common density-delay curve, supporting the predicted scaling behavior. The result provides a compact law for stable event-driven state progression in open distributed systems and offers a scaling-based interpretation of Bitcoin-style difficulty adjustment as a decentralized way to regulate effective event density.

Problem

• distributed systems
• event-driven state progression
• network delay
• fork depth
• stability

Innovation

• density-delay law
• event-driven state progression
• fork depth
• Poisson approximation
• inverse-scaling