Discontinuous transition in explosive percolation via local suppression

TL;DR

The paper investigates a dynamical explosive percolation process in which a newly added link triggers sequential rewiring of neighboring nodes to suppress giant-cluster formation, using only local information. The authors analyze a Bethe lattice branch and a bipartite network, showing that a discontinuous percolation transition can occur even when the number of rewiring nodes per addition, $m(p)$, remains finite on the branch and diverges only near the critical point on the bipartite network. The order parameter $P_{9inf}(p)$ (or $G(p)$ for the bipartite case) aligns with steady-state results, indicating that local rewiring can effectively realize a DPT without global information. This extends the understanding of EP by demonstrating that discontinous transitions can arise under local information plus dynamical rewiring, broadening the classes of networks and rules that yield DPTs.

Abstract

We study an explosive percolation model in which a link is randomly added and neighboring nodes sequentially rewire their links to suppress the growth of large clusters. In this manner, the rewiring nodes spread outward starting from the initial node closest to the added link. We show that a discontinuous transition emerges even when the total number of rewiring nodes after each link addition is finite. This finding implies that adding a link using the information of the cluster sizes attached to a finite set of link candidates (local information) can lead to a discontinuous transition if link rewiring is allowed. This result thus extends the previous result that a discontinuous transition arises only when a link is added using the information of the cluster sizes attached to an infinite number of link candidates (global information) in the absence of rewiring.

Figures (7)

• Figure 1: Schematic diagram of one link addition in the EP model on a branch with $L=4$. (a) Initial configuration, where solid lines denote occupied links and dotted lines denote unoccupied links. (b) One link (thick red solid line) is added to the initial rewiring node (empty circle) following steps (b1) and (b2) given in Sec. \ref{['sec:branch']}. (c) The node (empty circle) reached via the occupied link from below rewires its links in ascending order of neighboring cluster sizes. (d) The link to the parent of the rewiring node in (c) is unoccupied, and thus the process of adding one link terminates.
• Figure 2: (a) $P_{\infty}(p)$ of the EP model (solid line) and the steady state (symbols) on the branch with $L=10$ ($\square$ with line) and $16$ ($\bigcirc$ with line). The dashed line is the theoretical curve of $P_{\infty}(p)$ in the thermodynamic limit $L \rightarrow \infty$. (b) $m$ vs. $p_c-p$ on the branch with $L = 10$ ($\square$) and $16$$(\circ)$. The dotted line indicates the estimated value $m(p_c) = 2.894$.
• Figure 3: (a) $G(t)$ for $N/10^3=16$ at $p=0.61$$(\square)$, $0.62$$(\blacksquare)$, $p_c$$(\circ)$, $0.63$$(\bullet)$, and $0.64$$(\triangle)$, shown from bottom to top. (b) $f(t)$ for $N/10^3=16$ at $p=0.61$$(\square)$, $0.62$$(\blacksquare)$, $p_c$$(\circ)$, $0.63$$(\bullet)$, and $0.64$$(\triangle)$. (c) $f(p)$ for $N/10^3=1$$(\circ)$, $4$$(\blacksquare)$, and $16$$(\square)$.
• Figure 4: Schematic diagram of link addition in the EP model on a bipartite network, where the circles (squares) represent nodes in the first (second) partition, solid lines denote occupied links, and dotted lines denote unoccupied links. (a) One link (red line) is added to the initial rewiring node (empty circle) following steps (d1--d3). (b) Following the solid arrows according to step (d4), the empty circle becomes the current rewiring node set. Then, it rewires according to steps (d2)--(d4). Since the next rewiring node set becomes empty in step (d5), the process terminates.
• Figure 5: Schematic diagram of the transient process following steps (d2--d4). (a) The circular node, approached via the upper arrow, rewires its link along the dotted arrow. The node reached via the bottom arrow becomes the next rewiring node. (b) The circular node reached via the upper arrow maintains its occupied links, and the node reached via the bottom arrow becomes the next rewiring node. (c) In the left panel, the circular node on the left rewires its link along the dotted arrow. In the right panel, the circular node on the right rewires its link to the common square node along the dotted arrow.