Continuity of the explosive percolation transition

Abstract

The explosive percolation problem on the complete graph is investigated via extensive numerical simulations. We obtain the cluster-size distribution at the moment when the cluster size heterogeneity becomes maximum. The distribution is found to be well described by the power-law form with the decay exponent τ = 2.06(2), followed by a hump. We then use the finite-size scaling method to make all the distributions at various system sizes up to N = 2 37 collapse perfectly onto a scaling curve characterized solely by the single exponent τ . We also observe that the instant of that collapse converges to a well-defined percolation threshold from below as N → ∞. Based on these observations, we show that the explosive percolation transition in the model should be continuous, contrary to the widely-spread belief of its discontinuity.PACS numbers: 64.60.ah, 64.60.aq, 36.40.Ei The term explosive percolation was proposed in Ref.[1] to describe a sudden appearance of a macroscopic cluster in a network growth model with the so-called product rule considered on the complete graph. This growth rule, named as the Achlioptas process (AP), is then studied on the two-dimensional lattice [2,3] and on the scalefree networks [4][5][6] as well, yielding similar results. That suddenness has been widely believed to indicate a discontinuity at the percolation transition in the thermodynamic limit [7,8], and the similar explosiveness has been observed with the other growth rules proposed later [9][10][11][12][13]. These observations of the explosiveness have drawn much interest due to the striking difference from the wellknown continuous transition in the standard percolation models [14]. However, in our point of view, the explosiveness has not been carefully investigated as yet enough to draw a decisive conclusion on the discontinuity, and possibly just represents an extremely steep but still continuous transition.Friedman and Landsberg [9] have suggested the argument of the powder keg as a circumstantial description to explain the apparent discontinuity of the explosive percolation transition. Meanwhile, da Costa et al. [15] have reported that the explosive percolation is actually continuous for a modified version of the AP by analytically deriving the critical scaling relations based on numerical observations of power-law critical distribution of cluster size [16]. In this Letter, we try to unmask the (dis)continuity in a systematic and direct way by performing a careful finite-size-scaling analysis at newly introduced pseudo-transition points for finite systems and show that the explosive percolation transition on the complete graph is indeed continuous in the thermodynamic limit.The model we study is the AP with the product rule on the complete graph [1]. Start with N nodes with all links unoccupied. At each step, choose two possible unoccupied links randomly between nodes. Then, select the link merging two clusters with a smaller product of the two cluster sizes. Here, a cluster is defined as a set of nodes connected each other via...

Explosive percolation on the Bethe lattice

“…By applying SCS to the well-known percolation models, random bond percolation and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing the key functions obtained from SCSs for the Achlioptas processes (APs) with a product rule and a sum rule to the prototype functions, we show that the percolation transition of AP models on the Bethe lattice is continuous regardless of details of growth rules.-- [2][3][4][5][6][7][8][9][10][11][12]. Achlioptas process (AP) was originally argued to show the discontinuous phase transition on the complete graph (CG) by suppressing growth of large clusters [1].…”

“…-- [2][3][4][5][6][7][8][9][10][11][12]. Achlioptas process (AP) was originally argued to show the discontinuous phase transition on the complete graph (CG) by suppressing growth of large clusters [1].…”

“…In contrast Riordan and Warnke [13] analytically showed that the phase transition in AP model [1] on CG is continuous by use of the arbitrary connectivity of CG. Furthermore several studies also showed that the transition of variants of EP models on CG is continuous [10][11][12]. However, EP on CG was still reported to undergo discontinuous transition depending on the detail of cluster growth rule [14,15].…”

Agglomerative percolation on the Bethe lattice and the triangular cactus

“…But subsequent studies on the explosive percolation have shown that the transition of the explosive percolation on CG or the mean-field transition is continuous [4][5][6][7].…”

“…Recently the complete graph is widely used as a testbed for MFT [2,[4][5][6]. However the complete graph is not bipartite and one growth step of AP on the complete graph makes the entire graph a new single cluster.…”