Information entropy of classical versus explosive percolation

Percolation is used to model diverse phenomena, ranging from porous media to social interactions. There is a well-known phase transition associated with percolation, characterized by the emergence of a giant cluster at the critical point, comparable in size to the entire network. Besides the classical percolation, in which the giant cluster emerges from a completely random process of adding edges to a network, in recent years a novel type of percolation, called explosive, has been studied. In explosive percolation, the addition of edges includes a choice process aimed to retard the emergence of the giant cluster. Looking for a method of studying phase transitions of percolating systems without the explicit use of order parameter (relative size of the network's largest cluster), we analyze the Shannon entropy associated with the cluster size probability distribution. It's known that at the critical point the cluster size distribution is a power-law, i.e. there are clusters of all sizes, so one expects the information entropy to attain a maximum. As expected, our results show that the entropy attains a maximum at this point for classical percolation. Surprisingly, for explosive percolation the maximum entropy does not match the critical point. Moreover, we show that it is possible determine the critical point without using the conventional order parameter, just analysing the entropy's derivatives.