Understanding percolation phase transition behaviors in complex networks from the macro and meso-micro perspectives

Abstract

Over the most recent twenty years, network science has bloomed and impacted different fields such as statistical physics, computer science, sociology, and so on. Studying the percolation behavior of a network system has a very important role in vital nodes identification, ranking, network resilience, and propagation behavior of networks. When a network system undergoes failures, network connectivity is broken. In this perspective, the percolation behavior of the giant connected component and finite-size connected components is explored in depth from the macroscopic and meso-microscopic views, respectively. From a macro perspective, a single network system always shows second-order phase transitions, but for a coupled network system, it shows rich percolation behaviors for various coupling strength, coupling patterns and coupling mechanisms. Although the giant component accounts for a large proportion in the real system, it cannot be neglected that when the network scale is large enough, the scale of finite-size connected components has an important influence on network connectivity. We here systematically analyze the phase transition behaviors of finite-size connected components that are different from the giant component from a meso-microscopic perspective. Studying percolation behaviors from the macro and meso-micro perspectives is helpful for a comprehensive understanding of many fields of network science, such as time-series networks, adaptive networks, and higher-order networks. The intention of this paper is to provide a frontier research progress and promising research direction of network percolation from the two perspectives, as well as the essential theory of percolation transitions on a network system.