Complex systems are large collections of entities that organize themselves into non-trivial structures, represented as networks. One of their key emergent properties is robustness against random failures or targeted attacks —i.e., the networks maintain their integrity under removal of nodes or links. Here, we introduce network entanglement to study network robustness through a multiscale lens, encoded by the time required for information diffusion through the system. Our measure’s foundation lies upon a recently developed statistical field theory for information dynamics within interconnected systems. We show that at the smallest temporal scales, the node-network entanglement reduces to degree, whereas at extremely large scales, it measures the direct role played by each node in keeping the network connected. At the meso-scale, entanglement plays a more important role, measuring the importance of nodes for the transport properties of the system. We use entanglement as a centrality measure capturing the role played by nodes in keeping the overall diversity of the information flow. As an application, we study the disintegration of empirical social, biological and transportation systems, showing that the nodes central for information dynamics are also responsible for keeping the network integrated.

复杂系统是将自身组织成非平凡结构的大型实体集合，表示为网络。它们的关键紧急特性之一是对随机故障或针对性攻击的鲁棒性——即，网络在删除节点或链接的情况下保持其完整性。在这里，我们引入网络纠缠以通过多尺度镜头研究网络鲁棒性，该镜头由通过系统的信息传播所需的时间进行编码。我们测量的基础在于最近开发的用于互连系统内信息动态的统计场理论。我们表明，在最小的时间尺度上，节点网络纠缠减少到一定程度，而在极大的尺度上，它衡量每个节点在保持网络连接方面发挥的直接作用。在中尺度上，纠缠起着更重要的作用，衡量节点对系统传输特性的重要性。我们使用纠缠作为中心性度量来捕捉节点在保持信息流的整体多样性方面所扮演的角色。作为一个应用，我们研究了经验社会、生物和交通系统的瓦解，表明信息动态中心的节点也负责保持网络的整合。