Local Cluster Aggregation Models of Explosive Percolation

We introduce perhaps the simplest models of graph evolution with choice that demonstrate discontinuous percolation transitions and can be analyzed via mathematical evolution equations. These models are local, in the sense that at each step of the process one edge is selected from a small set of potential edges sharing common vertices and added to the graph. We show that the evolution can be accurately described by a system of differential equations and that such models exhibit the discontinuous emergence of the giant component. Yet they also obey scaling behaviors characteristic of continuous transitions, with scaling exponents that differ from the classic Erdős-Rényi model.