Criticality and scaling behavior of percolation with multiple giant clusters under an Achlioptas process

Achlioptas processes, a class of percolation models which can lead to rich critical phenomena, including the well-known explosive percolation, have attracted much attention in recent years. In this paper, we show that, in a three-vertex Achlioptas process, two giant clusters emerge after the percolation transition with size fluctuations in different realizations, and the choice of the connecting vertex in the smaller cluster depends on a probability parameter p, the increase of which can make the transition sharper. Using finite-size scaling analysis, we can determine the critical point r(c) and critical exponents η, 1/ν, and β through Monte Carlo simulations. Comparison of such exponents for different giant clusters indicates that their critical nature is always the same. However, when link choice is strongly biased, it is surprising that the scaling relation η=β/ν is violated, and the data collapse for scaling function diverges. Furthermore, by inspecting the variance of exponents with p, three distinct scaling phases are classified for different parameter intervals according to the divergence scaling function, which suggests an inconsistent scaling form in the critical window with the supercritical region. The study on the criticality and scaling behavior of multiple giant clusters in an Achlioptas process, in particular, the discovery of three scaling phases that depend on the parameter p, may help us in finding a complete scaling theory for the Achlioptas-process percolation and give insight into understanding the accelerating nature of the phase transition for Achlioptas processes once reaching criticality.