Explosive Percolation Transition is Actually Continuous

Recently a discontinuous percolation transition was reported in a new “explosive percolation” problem for irreversible systems [D. Achlioptas, R. M. D’Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.

Figure 1

(a) In classical percolation, at each step, two randomly chosen nodes are connected by a new link. If these nodes belong to different clusters, these clusters merge. (b) In the explosive percolation model, at each step, two pairs of nodes are chosen at random, and for each of the pairs, the node belonging to the minimal cluster is selected. These two nodes (and so their clusters) are connected by a new link. (c) The relative size $S$ of a percolation cluster versus $t$ for explosive percolation obtained by simulation of the model with $2\times {10}^9$ nodes (1000 runs) and, for comparison, $S(t)$ for classical percolation. Inset: the log-log plot $S$ vs $t$ for this data.

Figure 2

Log-log plot $S$ vs $t-t_c$ obtained by solving Eq. (2).

Figure 3

Solutions of Eq. (2) for the infinite system. (a) The evolution of the distribution $P(s)$ below and at (solid lines) and above (dashed lines) the percolation threshold for explosive percolation. (b) The evolution of $P(s)$ for ordinary (classical) percolation. (c) Dependence of $P(s,t)$ on $t$ for a set of cluster sizes $s$ for explosive percolation. Numbers on curves indicate $s$. (d) $P(s,t)$ versus for normal percolation. The insets show the $P(s,t)$ curves for large values of $s$.

Figure 4

Scaling functions $f(x)$ and $g(x)$ for explosive percolation for $t<t_c$ (left) and $t>t_c$ (right), and, for comparison, the scaling function $f_{\mathrm{CP}}(x)$ for ordinary percolation.