Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, 90007, USA.
Sci Rep. 2021 Nov 25;11(1):22964. doi: 10.1038/s41598-021-02203-4.
Network theory helps us understand, analyze, model, and design various complex systems. Complex networks encode the complex topology and structural interactions of various systems in nature. To mine the multiscale coupling, heterogeneity, and complexity of natural and technological systems, we need expressive and rigorous mathematical tools that can help us understand the growth, topology, dynamics, multiscale structures, and functionalities of complex networks and their interrelationships. Towards this end, we construct the node-based fractal dimension (NFD) and the node-based multifractal analysis (NMFA) framework to reveal the generating rules and quantify the scale-dependent topology and multifractal features of a dynamic complex network. We propose novel indicators for measuring the degree of complexity, heterogeneity, and asymmetry of network structures, as well as the structure distance between networks. This formalism provides new insights on learning the energy and phase transitions in the networked systems and can help us understand the multiple generating mechanisms governing the network evolution.
网络理论帮助我们理解、分析、建模和设计各种复杂系统。复杂网络编码了自然界中各种系统的复杂拓扑结构和结构相互作用。为了挖掘自然和技术系统的多尺度耦合、异质性和复杂性，我们需要表达性和严格的数学工具，这些工具可以帮助我们理解复杂网络及其相互关系的生长、拓扑、动力学、多尺度结构和功能。为此，我们构建了基于节点的分形维数（NFD）和基于节点的多重分形分析（NMFA）框架，以揭示动态复杂网络的生成规则，并量化其依赖于尺度的拓扑和多重分形特征。我们提出了新的指标来衡量网络结构的复杂性、异质性和不对称性，以及网络之间的结构距离。这种形式主义为学习网络系统中的能量和相变提供了新的见解，并帮助我们理解控制网络演化的多种生成机制。
