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

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.

💡 Research Summary

The paper addresses a fundamental scaling problem in open, permission‑less distributed systems where state updates are mutually exclusive: how many proposals can the system sustain without the fork depth growing without bound when network propagation delay is non‑negligible? The authors model the system as follows. A set of N nodes independently generate candidate state transitions at a per‑node attempt rate g(N). The aggregate proposal intensity is λ = N·g(N). Because a proposal needs a finite time Δ to become visible to all participants, each proposal occupies a “propagation window” of length Δ. Any other proposal that arrives within this window overlaps with the first one, creating a conflict (fork).

Assuming proposal arrivals follow a Poisson process, the number of proposals KΔ that fall inside a window of length Δ is Poisson‑distributed with mean λΔ. The collision probability is therefore

 P_coll = 1 – e^{‑λΔ}(1 + λΔ).

When λΔ ≪ 1, the approximation P_coll ≈ (λΔ)²/2 holds, showing that collisions are a second‑order effect in the density‑delay product.

Fork dynamics are captured by a birth‑death process. At each discrete decision epoch τ, a collision (with probability P_coll) increases the fork depth Fτ by one, while a resolution event (with probability q) decreases it by one. The expected drift is

 E