Scaling behavior of information entropy in explosive percolation transitions

An explosive percolation transition is the abrupt emergence of a giant cluster at a threshold caused by a suppression of the growth of large clusters. In this paper, we consider the information entropy of the cluster-size distribution, which is the probability distribution for the size of a randomly chosen cluster. It has been reported that information entropy does not reach its maximum at the threshold in explosive percolation models, a result seemingly contrary to other previous results that the cluster-size distribution shows power-law behavior and the cluster-size diversity (number of distinct cluster sizes) is maximum at the threshold. Here, we show that this phenomenon is due to the fact that the scaling form of the cluster-size distribution is given differently below and above the threshold. We also establish the scaling behaviors of the first and second derivatives of the information entropy near the threshold to explain why the first derivative has a negative minimum at the threshold and the second derivative diverges negatively (positively) at the left (right) limit of the threshold, as predicted through previous simulation.