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

TL;DR

This study investigates phase transitions in diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks, revealing various transition types including first, second, and third-order transitions.

Contribution

It provides a detailed analysis of percolation thresholds and transition orders, including the novel discovery of third-order transitions in bootstrap percolation with m=2.

Findings

Diffusion percolation with small k exhibits double transitions.

Bootstrap percolation with m≥3 shows first-order transition.

Third-order transition found in bootstrap percolation with m=2.

Abstract

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.