  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.
