The Achlioptas process, a percolation algorithm on random network, shows a rapid second-order phase transition referred to as explosive percolation. To obtain the transition point and critical exponent $\beta$ for percolations on a random network, especially for bond percolations, we propose a new scaling analysis that is independent of the system size. The transition point and critical exponent $\beta$ are estimated for the product-rule (PR) and da Costa-Dorogovtsev-Goltsev-Mendes (dCDGM) ($m=2$) models of the Achlioptas process, as well as for the Erdős-Rényi (ER) model, which is a classical model in which the analytic values are known. The validity of the scaling analysis is confirmed, especially for the transition point. The estimations of $\beta$ are also consistent with previously reported values for the ER and dCDGM(2) models, whereas the $\beta$ estimation for the PR model deviates somewhat. By introducing a parameter representing the maximum cluster size, we develop an extrapolation scheme for the critical exponent $\beta$ from the simulation just at the transition point in order to obtain a more accurate value. The estimated value of $\beta$ is improved compared with that obtained by the scaling analysis for the ER model and is also consistent with the $\beta$ value obtained for the dCDGM(2) model, whereas its deviation from the previously reported value is larger for the PR model. We discuss the accuracy of the present estimations and draw conclusions about their reliability.

• Received 14 April 2022
• Accepted 31 October 2022

DOI:https://doi.org/10.1103/PhysRevE.106.054138