Inverting the Achlioptas rule for explosive percolation

TL;DR

This paper investigates a novel inverse Achlioptas process where the largest clusters merge, leading to a continuous percolation transition with distinct scaling properties, contrasting the explosive transition seen in the traditional process.

Contribution

It introduces and analyzes a new inverse Achlioptas process, developing a theoretical framework and characterizing its continuous percolation transition.

Findings

The inverse process results in a continuous transition similar to ordinary percolation.

The critical exponents and scaling functions are derived for this process.

The transition occurs in less connected systems compared to standard percolation.

Abstract

In the usual Achlioptas processes the smallest clusters of a few randomly chosen ones are selected to merge together at each step. The resulting aggregation process leads to the delayed birth of a giant cluster and the so-called explosive percolation transition showing a set of anomalous features. We explore a process with the opposite selection rule, in which the biggest clusters of the randomly chosen ones merge together. We develop a theory of this kind of percolation based on the Smoluchowski equation, find the percolation threshold, and describe the scaling properties of this continuous transition, namely, the critical exponents and amplitudes, and scaling functions. We show that, qualitatively, this transition is similar to the ordinary percolation one, though occurring in less connected systems.