Deciphering the generating rules and functionalities of complex networks.

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.