Phase transition behavior of finite clusters under localized attack

Most previous studies focused on the giant component to explore the structural robustness of complex networks under malicious attacks. As an important failure mode, localized attacks (LA) can excellently describe the local failure diffusion mechanism of many real scenarios. However, the phase transition behavior of finite clusters, as important network components, has not been clearly understood yet under LA. Here, we develop a percolation framework to theoretically and simulatively study the phase transition behavior of functional nodes belonging to the finite clusters of size greater than or equal to s ( s = 2 , 3 , horizontal ellipsis ) under LA in this paper. The results reveal that random network exhibits second-order phase transition behavior, the critical threshold p c increases significantly with increasing s, and the network becomes vulnerable. In particular, we find a new general scaling relationship with the critical exponent delta = - 2 between the fraction of finite clusters and s. Furthermore, we apply the theoretical framework to some real networks and predict the phase transition behavior of finite clusters in real networks after they face LA. The framework and results presented in this paper are helpful to promote the design of more critical infrastructures and inspire new insights into studying phase transition behaviors for finite clusters in the network.