Abstract
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.
Highlights
A multifractal asymmetry metric to measure the multiscale structural asymmetry of a network
We propose the node-based fractal dimension (NFD) and the node-based multifractal analysis (NMFA) framework to decipher the topological information and the generating rules embedded in the network structures, based on which we define a general approach to quantify the complexity and heterogeneity of networks
We propose two novel indicators to measure the structure distance between networks and the multiscale asymmetry of a network structure, which could prove revolutionary in the field of network analysis
Summary
A multifractal asymmetry metric to measure the multiscale structural asymmetry of a network. We show that the NMFA framework can decipher the generating rules of networks and reveal the relationship between network structures and functionalities. The NMFA framework proves to be a powerful and stable tool to investigate a variety of networks and helps us comprehensively learn their topological features and higher-order connectivity patterns, as well as reveal the structure, dynamics, generating rules and functionalities of the network and their interrelationship, opening a new chapter in network science
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