发表机构
Center for Science of Science and Innovation, Kellogg School of Management, Northwestern University; Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University; Université Paris-Saclay; CEA; CNRS; Institut de Physique Théorique; Centre d’Analyse et de Mathématique Sociales (CNRS/EHESS); Complexity Science Hub, Vienna（西北大学凯洛格管理学院科学科学与创新中心; 印第安纳大学卢迪信息学、计算与工程学院复杂网络与系统研究中心; 巴黎萨克雷大学; 法国原子能和替代能源委员会; 法国国家科学研究中心; 理论物理研究所; 社会分析与数学中心（CNRS/EHESS）; 维也纳复杂性科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究基于全球19个都市约3000条公交路线数据，揭示公交频率随需求的标度规律，该规律源于城市在固定运营预算下最小化乘客总等待时间的优化原则，存在拥挤可忽略和过载两种普适 regimes，还发现额外投资收益在不同系统间不均。
AI 中文摘要
城市必须分配有限资源以维持流动性，却对系统最终状态存在不确定性。通过分析全球19个都市约3000条年客运量超40亿人次的公交路线，我们揭示了形式为$f \sim (d/t)^\alpha$、指数为$\alpha \in [1/2,\,2/3]$的稳健标度律，该定律将服务频率$f$与乘客需求$d$、路线时长$t$关联起来。我们证明，该标度源于一个简单优化原则：当同时考虑发车间隔和拥挤程度时，城市会在固定运营预算下隐性最小化乘客总等待时间。该机制产生两种普适 regime：拥挤可忽略时为频率主导 regime，对应$\alpha = 1/2$；多数路线过载时为运力主导 regime，对应$\alpha = 2/3$。当仅部分网络接近运力时，会出现中间指数。此外，我们发现额外投资的收益在不同系统间高度不均。例如，我们的模型显示，$20\%$的预算增加会使波士顿每位乘客的每日等待时间减少近5分钟，而巴黎仅减少约1分钟。这些发现将城市公共交通归入更广泛的受限运力分配问题类别，同时凸显了一种独特 regime，其中规定的路线需求决定了有限服务资源的分配。所得标度律表明，简单优化原则如何在复杂交通系统中产生系统性指数，超出了物理和生物流动网络中通常考虑的基于耗散的框架。
英文摘要
Cities must allocate limited resources to maintain mobility, with uncertainties about the resulting state of the system. Analyzing roughly 3,000 bus routes with more than 4 billion yearly riders across 19 metropolitan areas worldwide, we uncover a robust scaling law of the form $f \sim (d/t)^α$ with exponent $α\in [1/2,\,2/3]$, linking the service frequency $f$ to passenger demand $d$ and route duration $t$. We show that this scaling emerges from a simple optimization principle: cities implicitly minimize total passenger waiting time under a fixed operational budget when both schedule frequency and crowding are taken into account. This mechanism produces two universal regimes: a frequency-dominated regime with $α= 1/2$ when crowding is negligible, and a capacity-dominated regime with $α= 2/3$ when most routes are overloaded. Intermediate exponents arise when only part of the network operates near capacity. Furthermore, we find that the benefits of additional investment are highly uneven across systems. For instance, our model suggests that a $20\%$ budget increase yields nearly a 5-minute reduction in daily waiting time per passenger in Boston, compared to only about 1 minute in Paris. These findings place urban transit within a broader class of constrained capacity-allocation problems, while highlighting a distinct regime in which prescribed route demands shape the allocation of limited service resources. The resulting scaling laws show how simple optimization principles can generate systematic exponents in complex transport systems, beyond the dissipation-based frameworks usually considered in physical and biological flow networks.
Journal refProc. Natl. Acad. Sci. U.S.A. 123 (29) e2535998123 (2026)
