Discontinuous percolation via suppression of neighboring clusters in a network

Abstract

Our recent study on the Bethe lattice reported that a discontinuous percolation transition emerges as the number of occupied links increases and each node rewires its links to locally suppress the growth of neighboring clusters. However, since the Bethe lattice is a tree, a macroscopic cluster forms as an infinite spanning tree but does not contain a finite fraction of the nodes. In this paper, we study a bipartite network that can be regarded as a locally tree-like structure with long-range neighbors. In this network, each node in one of the two partitions is allowed to rewire its links to nodes in the other partition to suppress the growth of neighboring clusters. We observe a discontinuous percolation transition characterized by the emergence of a single macroscopic cluster containing a finite fraction of nodes, followed by critical behavior of the cluster size distribution. We also provide an analytical explanation of the underlying mechanism.

Figures (8)

• Figure 1: Bipartite network with $z=3$ and $N=8$ divided into two partitions, represented as the top and bottom rows. Solid lines represent occupied links, while dotted lines indicate unoccupied links. A node ($\bigcirc$) with $n=2$ occupied links is randomly selected from the bottom row. After the node disconnects all of its links, its three neighbors (from left to right in the top row) belong to clusters of sizes three, two, and one, respectively. To occupy $n$ links in ascending order of neighbor cluster sizes, the node occupies links to the two rightmost neighbors. As a result of this process, the location of the occupied link changes as indicated by the arrow.
• Figure 2: $G(p)$ for a single realization with $N/10^3 = 1$ ($\circ$), $8$ ($\bullet$), and $64$ ($\square$), shown for (a) $z = 3$ and (b) $z = 4$.
• Figure 3: Comparison of $1-(1-pP_{\infty}(p))^3(\bigcirc)$ and $Q_{\infty}(p)(\blacksquare)$ from a single realization for each $p$, with $N=16000$ and $z=3$. To obtain $P_{\infty}$ and $Q_{\infty}$, we computed the fraction of nodes belonging to the largest cluster in the first and second partitions, respectively, at steady state. The vertical dotted line indicates $p_c$.
• Figure 4: (a) Plots of $f(R_{\infty}) - g(R_{\infty})$ for $p = 0.61$, $p = p_c$, and $p = 0.64$, from bottom to top, with $z = 3$. (b) $G(p)$ for $N/10^3 = 1$ ($\circ$), $8$ ($\bullet$), and $64$ ($\square$) with $z = 3$. The dashed line represents the theoretical curve that describes the behavior of $G(p)$ near $p_c$.
• Figure 5: Plots of $n_s$ for $p=0.1 (\square)$, $0.3 (\blacksquare)$, $0.5 (\bigcirc)$, and $0.63 (\bullet)$ with $N=16000$. The solid lines represent the theoretical curves for $p = 0.1$, $0.3$, $0.5$, and $p_c$, from left to right. We note that the curve for $n_s$ at $p = 0.63>p_c$ aligns well with the theoretical prediction for $p_c$. This is due to finite-size effects, whereby the value of $p$ at which $n_s$ exhibits a power-law behavior decreases to $p_c$ as $N$ increases PREhybrid2D. Inset: Theoretical curve for $p=p_c$ (solid line), extended over a wide range of $s$. The slope of the dashed line is $-3$, confirming the estimated critical exponent $\tau=3$.