标题： 非平衡网络中的动态相变 显示英文标题

标题： Dynamical Phase Transitions in Non-equilibrium Networks

摘要： 动态相变（DPTs）表征系统行为在有限时间内发生的临界变化，为研究超越传统平衡物理的非平衡现象提供了一个视角。 尽管在量子系统中得到了广泛研究，DPTs 在经典环境中仍 largely 未被探索。 对从社交网络到金融市场的复杂系统最近的实验揭示了类似于量子 DPTs 的突然动态变化，促使人们寻找理论上的理解。 在这里，我们提出一个非平衡网络的最小模型，证明网络边之间的非线性相互作用自然导致 DPTs。 具体而言，我们表明网络度在有限的临界时间发散，遵循普遍的双曲标度，与实证观察一致。 我们的分析结果预测，关键网络属性，包括度分布和聚类系数，在临界性接近时表现出临界标度。 这些发现为理解跨多种复杂系统的涌现非平衡临界性建立了理论基础。 显示英文摘要

摘要： Dynamical phase transitions (DPTs) characterize critical changes in system behavior occurring at finite times, providing a lens to study nonequilibrium phenomena beyond conventional equilibrium physics. While extensively studied in quantum systems, DPTs have remained largely unexplored in classical settings. Recent experiments on complex systems, from social networks to financial markets, have revealed abrupt dynamical changes analogous to quantum DPTs, motivating the search for a theoretical understanding. Here, we present a minimal model for nonequilibrium networks, demonstrating that nonlinear interactions among network edges naturally give rise to DPTs. Specifically, we show that network degree diverges at a finite critical time, following a universal hyperbolic scaling, consistent with empirical observations. Our analytical results predict that key network properties, including degree distributions and clustering coefficients, exhibit critical scaling as criticality approaches. These findings establish a theoretical foundation for understanding emergent nonequilibrium criticality across diverse complex systems.