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Eur. Phys. J. B (2015) 88: 213DOI: 10.1140/epjb/e2015-60500-0Regular ArticleT HE E UROPEANP HYSICAL J OURNAL BInformation entropy of classical versus explosive percolationTiago M. Vieira 1,a , Gandhi M. Viswanathan 1,2 , and Luciano R. da Silva 1,21Departamento de F´?sica, Universidade Federal do Rio Grande do Norte, 59078-970 Natal, Rio Grande do Norte, Brazil2National Institute of Science and Technology of Complex Systems, Universidade Federal do Rio Grande do Norte,59078-970 Natal, Rio Grande do Norte, BrazilReceived 28 April 2015 / Received in final form 24 June 2015Published online 2 September 2015 – c ? EDP Sciences, Societ` a Italiana di Fisica, Springer-Verlag 2015Abstract. We study the Shannon entropy of the cluster size distribution in classical as well as explosivepercolation, in order to estimate the uncertainty in the sizes of randomly chosen clusters. At the criticalpoint the cluster size distribution is a power-law, i.e. there are clusters of all sizes, so one expects theinformation entropy to attain a maximum. As expected, our results show that the entropy attains amaximum at this point for classical percolation. Surprisingly, for explosive percolation the maximumentropy does not match the critical point. Moreover, we show that it is possible to determine the criticalpoint without using the conventional order parameter, just analysing the entropy’s derivatives.1 IntroductionPercolation [1,2] is used to model diverse phenomena,ranging from porous media to social interactions. Thereis a well-known phase transition associated with perco-lation. At the critical point, the giant cluster emerges,comparable in size to the entire network. The classicalnetwork model developed by Erdös and R´ enyi [3] (ran-dom network) has a smooth continuous phase transition,as shown in Figure 1. In this model edges are randomlyarranged between the nodes in the network. It was be-lieved until recently that all percolation transitions werecontinuous. However, the discovery of explosive percola-tion [4] led to questions regarding the continuous natureof the percolation phase transition [5–12].The key idea behind explosive percolation is to addedges in a manner to delay the onset of the percolationtransition. A specific choice is made before the addition ofeach new edge. This choice process has the aim of delay-ing the formation of the giant cluster. As a consequence, itgrows very suddenly at birth, leading to an abrupt phasetransition. The first proposed mechanism was called theproduct rule (PR) [4], which works as follows. Two nodepairs are selected and an edge is placed between the pairwhose product of the number of nodes in the connectingclusters is the smallest. The explosive percolation causedby the PR is shown in Figure 1. Actually researches agreethat the explosive percolation transition is continuous, butwith unusual characteristics [12]. In fact, explosive perco-lation is not yet completely understood and continues tobe an area of intense research [13].Looking for a method of studying phase transitionsof percolating systems without the explicit use of orderae-mail: tiagotmv@gmail.com00.20.40.60.810 0.25 0.5 0.75 1.0 1.25 1.5fraction of nodes in the largest clusteredges / nodesclassicalPRFig. 1. Illustration of percolation phase transitions in ran-dom networks (non-lattice). Note the smooth growing curvefor the classical percolation and the step-shaped curve for theexplosive percolation (product rule – PR). The plot shows thefraction of nodes in the networks’ largest cluster vs. the den-sity of edges, i.e. the ratio between the number of edges andthe number of nodes. These results, and all others hereafter,except when it is explicitly mentioned, are based on simulationof 500 percolation transitions of each type over networks with3 × 10 5 nodes.parameter, we analyze the Shannon entropy associatedwith the cluster size probability distribution. By doing so,we note that it is possible to determine the critical pointthrough the entropy’s derivatives. In addition, we find apreviously unknown (and unexpected) property of explo-sive percolation: in contrast with classical percolation, itscritical point does not correspond to a maximum of theShannon entropy of the cluster size distribution.