  In the last two decades, network science has blossomed and influenced various
fields, such as statistical physics, computer science, biology and sociology,
from the perspective of the heterogeneous interaction patterns of components
composing the complex systems. As a paradigm for random and semi-random
connectivity, percolation model plays a key role in the development of network
science and its applications. On the one hand, the concepts and analytical
methods, such as the emergence of the giant cluster, the finite-size scaling,
and the mean-field method, which are intimately related to the percolation
theory, are employed to quantify and solve some core problems of networks. On
the other hand, the insights into the percolation theory also facilitate the
understanding of networked systems, such as robustness, epidemic spreading,
vital node identification, and community detection. Meanwhile, network science
also brings some new issues to the percolation theory itself, such as
percolation of strong heterogeneous systems, topological transition of networks
beyond pairwise interactions, and emergence of a giant cluster with mutual
connections. So far, the percolation theory has already percolated into the
researches of structure analysis and dynamic modeling in network science.
Understanding the percolation theory should help the study of many fields in
network science, including the still opening questions in the frontiers of
networks, such as networks beyond pairwise interactions, temporal networks, and
network of networks. The intention of this paper is to offer an overview of
these applications, as well as the basic theory of percolation transition on
network systems.
