一种新的复杂网络动力学分析框架
A novel framework to analyze complex network dynamics
摘要：
图论构成了一个广泛使用和建立的领域，为复杂网络的表征提供了强大的工具。网络的复杂拓扑结构也可以通过在自维持振荡（同步模式）或类似传播的过程（如随机行走）的相互作用中观察到的集体动力学来研究。然而，网络往往是从真实的数据推断形成动态系统，这是不同于那些用来揭示其拓扑特征。这强调了一个理论框架的必要性，致力于复杂网络的结构和动力学之间的相互关系，作为同一枚硬币的两面。在这里，我们提出了一个严格的框架，基于网络响应随时间的变化（即，格林函数）来研究节点之间随时间的相互作用。为此，我们定义了\emph{flow}，它描述了网络连接和外部输入之间的相互作用。这个多变量的措施涉及到图的可通信性和地图方程的概念。我们说明我们的理论使用多变量Ornstein-Uhlenbeck过程，它描述了稳定和非保守的动态，但形式主义可以适应其他本地动态的格林函数是已知的。我们提供了经典网络的例子，如小世界环和层次网络的应用程序。我们的理论定义了一个全面的框架，是规范相关的有向和加权网络，从而铺平了一条新的道路，以修改网络分析的标准，从节点之间的成对相互作用的网络，包括社区检测的全局属性。
英文摘要：
Graph theory constitutes a widely used and established field providing powerful tools for the characterization of complex networks. The intricate topology of networks can also be investigated by means of the collective dynamics observed in the interactions of self-sustained oscillations (synchronization patterns) or propagation-like processes such as random walks. However, networks are often inferred from real data forming dynamic systems, which are different from those employed to reveal their topological characteristics. This stresses the necessity for a theoretical framework dedicated to the mutual relationship between the structure and dynamics in complex networks, as the two sides of the same coin. Here we propose a rigorous framework based on the network response over time (i.e., Green function) to study interactions between nodes across time. For this purpose we define the \emph{flow} that describes the interplay between the network connectivity and external inputs. This multivariate measure relates to the concepts of graph communicability and the map equation. We illustrate our theory using the multivariate Ornstein-Uhlenbeck process, which describes stable and non-conservative dynamics, but the formalism can be adapted to other local dynamics for which the Green function is known. We provide applications to classical network examples, such as small-world ring and hierarchical networks. Our theory defines a comprehensive framework that is canonically related to directed and weighted networks, thus paving a new way to revise the standards for network analysis, from the pairwise interactions between nodes to the global properties of networks including community detection.
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。
