Percolation processes that show a discontinuous, or at least very steep phase transition. See this image for a nice sumary of types of explosive percolation processes. The reviews below also summarize results, and below we discuss some of the main types.

Explosive Percolation: Novel critical and supercritical phenomena

Impact of single links in competitive percolation

Achlioptas processes

Achlioptas processes follow $m$-edge rules which involve choosing $m$ candidate edges uniformly at random between any pair of nodes (compare with other Spanning cluster-avoiding process)and applying a rule to select which one is actually chosen. These have been proven to be continuous in the thermodynamic limit, for a fixed $m$

k-vertex rule percolation process

Processes based on chosing $k$ vertices at random and adding edges among those vertices according to some rule. $k$-vertex rules are actually a generalization of $m$-edge rules.

Half-restricted processes

Half-restricted process is a variant of the Erdős–Rényi process which exhibits a discontinuous phase transition.

Explosive Percolation in Erdős-Rényi-Like Random Graph Processes

In each step, two vertices are connected by an edge, but one of them is restricted to be within the smaller components (more specifically defined to be a set composed of a given fraction, $f$, of the total nodes chosen in ascending order of {the size of the component they belong to}. This is also called the restricted vertex set, $R_f(G)$). note that the restricted vertex set is recalculated after every step, as the clusters have changed.

This process exhibits a discontinuous percolation transition for any $f < 1$

Spanning cluster-avoiding process

An spanning cluster-avoiding process (SCA) is an Explosive percolation model based on classifying bonds between those that facilitate the creation of the spanning-cluster, and those that don't, and preferentially selecting those that don't. They are similar to Achlioptas processes ($m$-edge processes). However, they don't require the candidate edges to be chosen at random between any pair of nodes, and instead the candidate edges can belong to a predetermined underlying network, common a hypercubic lattice. They are capable of showing discontinuous transitions, for certain choices of the number of candidate edges chosen per step

I think there should be a term used for $m$-edge-like processes, that have an underlying network..

Applications of explosive percolation models

• Jamming on the Internet. Congestion phenomena on complex networks.
• Synchronization phenomena. Explosive transitions to synchronization in networks of phase oscillators – Explosive Synchronization Transitions in Scale-Free Networks
• Analysis of real-world networks. Using explosive percolation in analysis of real-world networks