稀疏有向网络上复杂系统的动力学平均场理论
Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks
AI总结：
本文通过推广动力学平均场理论至稀疏有向网络，推导出单自由度有效动力学的精确路径概率方程，并利用种群动力学算法求解，揭示了神经网络模型随网络拓扑从不动点相到混沌的转变。
摘要：
尽管现实世界中的复杂系统通常通过稀疏且异质的网络进行交互，但其动力学的解析解仅限于具有全对全相互作用的模型。本文中，我们解决了具有随机结构的稀疏有向网络上广泛非线性复杂系统模型的动力学问题。通过将动力学平均场理论推广到稀疏系统，我们推导出一个描述单个自由度有效动力学的路径概率的精确方程。我们的通用解适用于神经网络、生态系统、流行病传播和同步研究中的关键模型。利用种群动力学算法，我们求解了路径概率方程，以确定一个开创性神经网络模型在稀疏区域的相图，表明该模型经历从不动点相到混沌的转变，该转变取决于网络拓扑结构。
英文摘要：
Although real-world complex systems typically interact through sparse and heterogeneous networks, analytic solutions of their dynamics are limited to models with all-to-all interactions. Here, we solve the dynamics of a broad range of nonlinear models of complex systems on sparse directed networks with a random structure. By generalizing dynamical mean-field theory to sparse systems, we derive an exact equation for the path-probability describing the effective dynamics of a single degree of freedom. Our general solution applies to key models in the study of neural networks, ecosystems, epidemic spreading, and synchronization. Using the population dynamics algorithm, we solve the path-probability equation to determine the phase diagram of a seminal neural network model in the sparse regime, showing that this model undergoes a transition from a fixed-point phase to chaos as a function of the network topology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
