arXiv:2606.01963 · v4

Entropy Bounds for Local Coordination and Graph Amenability

首次提交 2026-06-01 17:26

版本更新 2026-11-02 20:20

雷达观测 2026-11-04 23:42

摘要

We study local coordination on finite graphs, equivalently, the ability of randomized local protocols with independent private randomness to produce highly correlated outputs at neighboring vertices. Hutchcroft, Rospuskova, and Tamuz (2026) showed that if such local coordination can be achieved with low cost, then the graph admits a low-cost leader profile, or equivalently an appropriate bounded-radius decomposition, with a square-root loss in the cost. We substantially improve this bound for unbiased binary outputs. If the average disagreement is $\varepsilon$, we construct a leader profile of cost $O(\varepsilon\log(1/\varepsilon))$. The key ingredient is an information-theoretic influence measure: after fixing an ordering of the independent information sources, we assign to each binary output the increments of mutual information revealed along that ordering. For two outputs that disagree with probability $p$, the corresponding measures are within binary entropy $h(p)$ in total variation. Combined with a grand coupling theorem, this yields the improved bound. We also show that a universal linear bound is impossible. We construct graphs with binary local profiles of cost $O(\varepsilon)$ for which every leader profile has cost $$ Ω\!\left(\varepsilon\sqrt{\log(1/\varepsilon)}\right). $$ The construction uses weighted majority functions and can be implemented as an action-symmetric Nash equilibrium, giving a negative answer to the question of HRT whether leader equilibria are always optimal up to a constant factor. Finally, we show that the binary assumption is essential: for general real-valued outputs, the square-root dependence of the HRT theorem is sharp up to a universal constant.