Information entropy of classical versus explosive percolation

Abstract

We study the Shannon entropy of the cluster size distribution in classical as well as explosive percolation, in order to estimate the uncertainty in the sizes of randomly chosen clusters. 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.

Scaling behavior of information entropy in explosive percolation transitions

“…Interestingly, in [4], it was reported that the maximum point of H(t) is equal to t c in the ER model, whereas that of H(t) is less than t c in the EP models, as shown here in Fig. 1(a) and (b).…”

“…Using dCR, we are also able to understand why Ḣ(t) is minimum at t = t c and why Ḧ(t) diverges at t = t c in EP models, as reported in [4]. The rest of this paper is organized as follows.…”

“…Apart from the giant cluster, the distribution of finite-sized clusters also changes as the link occupation probability increases. The uncertainty of the distribution of finite-sized clusters has been studied using percolation models in various aspects [4][5][6][7][8][9][10]. In some of these results, it has been reported that the cluster size diversity, or information entropy [11], of the cluster size distribution reaches its maximum value at the threshold in (ordinary) random percolation models [4][5][6][7][8].…”

“…Specifically, in [4], the information entropy of the cluster size distribution in the Erdös-Rényi (ER) model [12] and in explosive percolation (EP) models [13][14][15][16] was studied.…”

Universality class of explosive percolation in Barabási-Albert networks

“…In percolation, we regard each cluster as equivalent to message hence the sum cannot be over the size of the clusters rather over the individual cluster label. In fact, none of the existing probabilities which we know in percolation theory can be used to measure entropy although there have been some attempts 43,44 .…”