Suppression Effect on Explosive Percolation

Percolation transitions (PTs) of networks, leading to the formation of a macroscopic cluster, are conventionally considered to be continuous transitions. However, a modified version of the classical random graph model was introduced in which the growth of clusters was suppressed, and a PT occurs explosively at a delayed transition point. Whether the explosive PT is indeed discontinuous or continuous becomes controversial. Here, we show that the behavior of the explosive PT depends on detailed dynamic rules. Thus, when dynamic rules are designed to suppress the growth of all clusters, the discontinuity of the order parameter tends to a finite value as the system size increases, indicating that the explosive PT could be discontinuous.

Figure 1

Classification of types of edge candidate pairs. (i) Both candidates $e_1$ and $e_2$ are intercluster edges. (ii) $e_1$ is an intracluster edge and the other candidate $e_2$ is an intercluster edge. (iii) Both $e_1$ and $e_2$ are intracluster edges.

Figure 2

(a) The fractions of type (ii) and type (iii) potential edges as a function of time $t$ for different system sizes: $N/{10}^4=32$, 64, 128, and 256 from the top (bottom) in the small-$t$ (large-$t$) region. As $N$ increases, the fraction increases dramatically. Inset: For a given $N=1.024\times {10}^7$, the fractions of the type (i) (dotted line) and the types (ii) and (iii) (solid lines) as a function of $t$. They are compared with the giant-cluster size $G_N(t)$ (dashed line). The fraction of the types (ii) or (iii) behaves similarly to $G_N(t)$, indicating that counting for the effect by taking the intracluster edges becomes important as $t$ approaches $t_c$. (b) The fraction of the occurrences in which the sum of the sizes of one pair of clusters becomes larger than that of the size of the other pair, even though the product of the sizes of the former is smaller than that of the sizes of the latter. The dotted line represents $G_N(t)$. Inset: Solid lines represent the failure ratio on an enlarged scale for different system sizes $N/{10}^4=32$, 64, 128, and 256. For a larger system, the curve lies on the upper position in the small-$t$ region. The dashed line represents $G_N(t)$. Numerical data for (a) and (b) are obtained from the ERPR-B model.

Figure 3

$G_N(t_x)$ versus $N$ for the (a) ERPR-A, (b) ERPR-B, (c) ERPR-C, (d) ERSR-A, (e) ERSR-B, and (f) ERSR-C models. The slopes of each guideline is (a) $-0.06$, (b) $-0.05$, (c) $-0.05$, (d) $-0.02$, (e) 0, and (f) 0. Thus, $G_N(t_x)$ of ERSR-B and ERSR-C converge to finite values in the thermodynamic limit. The error bars represent the deviation of the cross points. Each data set is obtained by taking an average over about ${10}^{13}/N$ configurations. $G_N(t_x)$ of models B and C of both the ERPR and ERSR overlap in the large-$N$ region. The insets of (a)–(f) show the behaviors of $G_N(t)$ for different system sizes $N/{10}^4=32$, 64, 128, 256, 512, 1024, 2048, 4096, 8192, and 16384. (g) Plot of the slope of $G_N(t)$ at $t_x$ versus $N$ for the ERSR-B (□) and the ERSR-C (○) models. $d{G}_N(t_x)/dt$ increases according to a power law $\sim N^{0.5}$.

Figure 4

(a) Schematic illustration of the selection rule on the intracluster edge for the CDGM-B model. In the original CDGM model (CDGM-A), when two pairs of edges $e_1$ and $e_2$ are selected from the cluster sets $C_1$ and $C_2$ of sizes $s_{1a}<s_{1b}$ and $s_{2a}<s_{2b}$, an additional edge connects the two nodes in the clusters $C_{1a}$ and $C_{2a}$ (dotted line). In our modified model (CDMA-B) the two nodes in the clusters $C_{1b}$ and $C_{2b}$ are connected instead (solid line), because by choosing them, the cluster sizes in the system do not increase. Panels (b)–(d) show $G_N(t_x)$ versus $N$ for CDGM-A, CDGM-B, and CDGM-C models, respectively, and the slopes are $-0.03$, 0, and 0, respectively. $G_N(t_x)$ of the models B and C overlap in the large $N$ region. It may be reasonable to expect that the modified CDGM model (CDGM-B and C) shows a discontinuous PT.