Suppression effect on the Berezinskii-Kosterlitz-Thouless transition in growing networks

The percolation transition in growing networks can be of infinite order, following the Berezinskii-Kosterlitz-Thouless (BKT) transition. Examples can be found in diverse systems, including coauthorship networks and protein interaction networks. Here, we investigate how such an infinite-order percolation transition is changed by the global suppression (GS) effect. We find that the BKT infinite-order transition breaks down, but the features of infinite-order, second-order, and first-order transitions all emerge in a single framework. Owing to the GS effect, the transition point $p_c$ is delayed, below which the critical region is extended. The power-law behavior of the cluster size distribution reaches the state with the exponent $\tau =2$ at $p_c$, suggesting that the system has the maximum diversity of cluster sizes and a first-order percolation transition occurs at $p_c$.

Figure 1

Schematic plots of (a) the order parameter $G$ and (b) the susceptibility $\langle s\rangle$ vs $q$ for the GRN.

Figure 2

Schematic illustration of the $r$-GRN model. Nodes (represented by balls) in set $R$ are dark green, whereas those in $R^c$ are blue. Each column represents a cluster. We start with the system at time $t=9$, which contains five clusters of sizes $(1,1,2,2,4)$ displayed from the second column on the left to the right in (a). Thus, $N\left(9\right)=10$. Here, the control parameter $g$ is taken as $g=0.4$; thus, $\lceil gN\rceil =4$. The sets $R$ and $R^c$ contain four (dark green) and six (blue) nodes, respectively. (a) At time $t=10$, the leftmost red open node is newly added. $N\left(t\right)=11$, and $\lceil gN\rceil =5$. Next, two nodes, say, those belonging to the first and second dark green clusters of sizes (1,1), are selected, and they are merged with probability $p$, making a cluster of size two. The largest cluster size in $R$, denoted as $S_R$, remains two. (b) At time $t=11$, again a new node is added (the leftmost open red ball). The new node is selected and merges into the second cluster in $R$, generating a cluster of size three. Because $N\left(11\right)=12$, $\lceil gN\rceil =5$. The merged cluster of size three moves to $R^c$, whereas the cluster of size two in $R^c$ moves to $R$. (c) At time $t=12$, a new node is added. $N\left(12\right)=13$, and $\lceil gN\rceil =6$. Two nodes are selected, but they are not connected with probability $1-p$. (d) At time $t=13$, a new node is added. $N\left(13\right)=14$, and $\lceil gN\rceil =6$. The fifth cluster from the left in $R$ and the first cluster in $R^c$ merge and generate a cluster of size five that belongs to $R^c$. The cluster of size four then lies on the border between the two sets. In this case, one node belongs to $R$, and the other three belong to $R^c$. $S_R$ is reset to four.

Figure 3

Plot of (a) $G$ and (b) $1/\langle s\rangle$ as a function of $p$ for the $r$-GRN model with $g=0.2$. Symbols represent the numerical simulation results for system sizes $N={10}^4$ ($◯$), ${10}^5$ ($▵$), ${10}^6$ ($\square$), and ${10}^7$ ($\diamond$). Each data point was averaged over ${10}^3$ configurations. The solid (red) lines are analytic results. The two vertical dotted lines represent $p_b$ and $p_c$.

Figure 4

Cluster size distribution $n_s\left(p\right)$ as a function of $s$ in different $p$ regions. Three cases of $n_s\left(p\right)$ are distinguished for $g=0.4$. (a) For $p<p_b$, $n_s\left(p\right)$ asymptotically follows the power law $\sim s^{-\tau \left(p\right)}$ with $\tau >3$. The slope of the dotted guide line is $-3$. Solid lines are obtained for $p=0.472576\approx p_b$, 0.45, 0.4, 0.3, 0.2, and 0.1 from right to left. (b) For $p_b\le p<p_c$, in the small-cluster-size region, $n_s\left(p\right)$ decays exponentially and then exhibits power-law behavior with $2<\tau \le 3$. Solid (black), dashed (red), and dashed-dotted (blue) lines represent $n_s\left(p\right)$ for $p=0.472576$, 0.657, and 0.659 45, respectively. Two dotted lines are guide lines with slopes of $-2$ and $-3$. (c) For $p\ge p_c$, $n_s\left(p\right)$ for finite clusters shows exponentially decaying distributions. Solid curves represent $p=0.6596$, 0.7, 0.8, 0.9, and 1.0 from right to left. The dotted curve is an exponentially decaying guide curve.

Figure 5

Phase diagram of the $r$-GRN model. Two transition points, $p_b$ ($▵$) and $p_c$ ($◯$), are determined for various $g$. $n_s(p,g)$ decays following a power law with $\tau \left(p\right)>3$ in the infinite-order critical region and $2<\tau \left(p\right)<3$ in the second-order critical region. Thus, the mean cluster size is finite and diverges in those regions, respectively. As $g$ approaches one, the two transition points are closer and converge to the critical point of an infinite-order transition, represented by $\blacksquare$.