Data from: Nonlinear growth: an origin of hub organization in complex networks

Many real-world networks contain highly connected nodes called hubs. Hubs are often crucial for network function and spreading dynamics. However, classical models of how hubs originate during network development unrealistically assume that new nodes attain information about the connectivity (for example the degree) of existing nodes. Here, we introduce hub formation through nonlinear growth where the number of nodes generated at each stage increases over time and new nodes form connections independent of target node features. Our model reproduces variation in number of connections, hub occurrence time, and rich-club organization of networks ranging from protein–protein, neuronal and fibre tract brain networks to airline networks. Moreover, nonlinear growth gives a more generic representation of these networks compared with previous preferential attachment or duplication–divergence models. Overall, hub creation through nonlinear network expansion can serve as a benchmark model for studying the development of many real-world networks.

许多真实世界网络中都存在连接度极高的节点，这类节点被称为枢纽节点（hub）。枢纽节点通常对网络功能与传播动力学至关重要。然而，现有关于网络演化过程中枢纽节点起源的经典模型存在不符合现实的假设：即新节点能够获取现有节点的连接状态（例如度值）信息。本文提出一种基于非线性增长的枢纽节点形成模型：每个阶段生成的节点数量随时间递增，且新节点与目标节点建立连接时，无需依赖目标节点的特征信息。我们的模型能够复现多种真实网络的连接数、枢纽节点出现时间以及富俱乐部（rich-club）结构的变化特征，涵盖蛋白质相互作用网络、神经元网络、大脑纤维束网络以及航空网络等。此外，相较于此前的偏好连接（preferential attachment）或复制-发散（duplication–divergence）模型，基于非线性增长的模型能够更通用地刻画这类真实网络。总体而言，通过非线性网络扩张实现枢纽节点构建的模型，可作为研究众多真实网络演化过程的基准模型。