Explosive percolation on the Bethe lattice is ordinary

TL;DR

This paper demonstrates that explosive percolation on a Bethe lattice behaves like ordinary percolation due to finite size effects, challenging previous notions of its abrupt transition.

Contribution

It reveals that the apparent explosive percolation on a Bethe lattice is actually ordinary percolation when finite size effects are properly accounted for.

Findings

The critical exponent matches that of ordinary percolation.

Finite size effects cause the apparent explosiveness.

Adjusting for finite size effects shows no true explosive transition.

Abstract

The Achlioptas process, which suppresses the aggregation of large-sized clusters, can exhibit an explosive percolation (EP) where the order parameter emerges abruptly yet continuously in the thermodynamic limit. It is known that EP is accompanied by an abnormally small critical exponent of the order parameter. In this paper, we report that a novel type of EP occurs on a Bethe lattice, where the critical exponent of the order parameter is the same as in ordinary bond percolation based on numerical analysis. This is likely due to the property of a finite Bethe lattice that the number of sites on the surface with only one neighbor is extensive to the system size. To overcome this finite size effect, we consider an approximate size of the cluster that each site on the surface along its branch belongs to, and accordingly approximate the sizes of an extensive number of clusters during simulation. As a result, the Achlioptas process becomes ineffective and the order parameter behaves like that of ordinary percolation at the threshold. We support this result by measuring other critical exponents as well.