A Delay-to-Timescale Ratio Law for Coordination

Abstract

Decentralized coordination collapses past a communication-delay threshold, but a threshold in steps is only meaningful relative to a timescale. A companion phase-diagram study located the collapse and, by varying the coordination gain, tied it to the swarm's intrinsic convergence time. That leaves the complementary question: if we impose an external task timescale $\tau_{\text{task}}$, does the boundary move with it? We build a delay-coupled tracking task in a simplified 3-D kinematic simulator in which a goal jumps once every $\tau_{\text{task}}$ and only a $15\%$ informed minority observes it — the remaining $85\%$ must acquire it through the delayed peer graph — and sweep $\tau_{\text{task}} \in \{15, \dots, 600\}$ s against one-way delay $d \in \{0, \dots, 40\}$ steps for gossip-consensus and flocking. The collapse delay $d_c$ grows with $\tau_{\text{task}}$: in the regime where the task is the binding timescale ($\tau_{\text{task}} = 30$–$150$ s) the ratio $d_c/\tau_{\text{task}} \approx 0.047$ is constant, so the cliff is governed by delay relative to the task timescale. For large $\tau_{\text{task}}$ ($\geq 300$ s) $d_c$ saturates at $\approx 9$ steps — the swarm's own convergence time becomes the shorter, binding timescale and the intrinsic cliff of the companion study reappears. A no-communication reference sits at the floor for every $\tau_{\text{task}}$ and delay, and a fully-informed control ($100\%$ observers) is delay-insensitive: the cliff is present only when coordination must flow through the delayed graph, confirming it is a coordination effect rather than a tracking artifact. Together with the intrinsic-timescale result, this pins the boundary to $d / \min(\tau_{\text{task}}, \tau_{\text{int}})$. We scope the claim precisely: a simulation-based algorithmic result about coordination primitives under delay, not physical-device validation.

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