Node-targeted Multi-step Percolation on Networks

The percolation, as an important theoretical tool for studying the robustness of networks, is continuously improved and explored on both ordinary graphs and hypergraphs. Given that most existing studies treat percolation as a single random event and overlook the fact that the collapse of real-world networks is a targeted and gradual process, we propose a node-targeted multi-step percolation model. In this model, the network undergoes multiple rounds of percolation processes, with the input for each percolation being the residual network after the previous percolation ends. Additionally, during each round of percolation, the failure probability of each node is related to the structure of the current residual network. It is worth noting that we have proposed a theoretical framework based on generating functions to predict the changes in the giant connected components of the networks during the multi-step percolation process, which has satisfactory predictive performance when the network does not have extreme heavy-tailed characteristics. Furthermore, through tests on randomly generated ordinary graphs and hypergraphs, we find that networks with Poisson degree/hyperdegree distributions are often more vulnerable than those with power-law degree/hyperdegree distributions when facing the proposed node-targeted multi-step percolation process.