Phase transition in the diffusion and bootstrap percolation models on regular random and Erd\H{o}s-R\'{e}nyi networks

The diffusion and bootstrap percolation models were studied in regular random and Erd\H{o}s-R\'{e}nyi networks using the modified Newman-Ziff algorithms. We calculated the percolation threshold and the order parameter of the percolation transition (strength of the giant cluster) and its derivatives. The percolation transitions are classified by the results. The diffusion percolation with a small $k$ has a double transition, and the bootstrap percolation with $m\geq3$ has the first-order percolation transition. The diffusion percolation with a large $k$ and the bootstrap percolation with a small $m$ show the second-order percolation transition. Particularly, third-order percolation transitions were discovered in the bootstrap percolation of $m=2$ in regular random networks.