Suppression effect on explosive percolations

When a group of people unknown to each other meet and familiarize among themselves, over time they form a community on a macroscopic scale. This phenomenon can be understood in the context of percolation transition (PT) of networks, which takes place continuously in the classical random graph model. Recently, a modified model was introduced in which the formation of the community was suppressed. Then the PT occurs explosively at a delayed transition time. Whether the explosive PT is indeed discontinuous or continuous becomes controversial. Here we show that type of PT depends on a detailed dynamic rule. Thus, when the dynamic rule is designed to suppress the growth of overall clusters, then the explosive PT could be discontinuous.

💡 Research Summary

The paper investigates why explosive percolation (EP) transitions can appear either continuous or discontinuous, focusing on the role of the “suppression principle” (SP) in the underlying edge‑selection dynamics. Starting from the classic Erdős–Rényi random graph, where edges are added uniformly and the percolation transition is continuous, the authors consider the Achlioptas process, which selects one edge from a set of candidates in order to suppress the formation of a giant component. The original Achlioptas rule (minimizing the product or sum of the sizes of the two clusters that would be merged) was thought to produce a discontinuous EP, but later studies suggested the transition might be continuous, leading to a controversy.

To resolve this, the authors classify candidate edge pairs into three types: (i) both candidates are inter‑cluster edges, (ii) one candidate is intra‑cluster while the other is inter‑cluster, and (iii) both candidates are intra‑cluster. For each type they define two selection criteria: the product rule (choose the edge with the smaller product of the two cluster sizes) and the sum rule (choose the edge with the smaller sum). They then construct three families of models—A, B, and C—differing in how they treat intra‑cluster candidates.

Model A always computes the product (or sum) even when a candidate is intra‑cluster; consequently it may select an edge that actually enlarges a cluster, violating the SP. Model B selects an intra‑cluster edge outright (thus preventing any growth) and, when both candidates are intra‑cluster, picks one at random. Model C restricts the dynamics to inter‑cluster edges only, effectively ignoring intra‑cluster candidates.

The authors perform extensive Monte‑Carlo simulations up to N = 10¹⁰ vertices, averaging over about 10¹³/N configurations. They locate the finite‑size crossing point tₓ(N) where the giant‑cluster fraction G_N(t) for size N intersects that for size 2N, and record G_N(tₓ). Two criteria are introduced to diagnose a discontinuous transition: (α) the pair (tₓ(N), G_N(tₓ)) remains finite as N→∞ (i.e., a non‑zero jump), and (β) the slope dG_N/dt at tₓ diverges with N.

Results for the product rule show that all three models (A, B, C) have G_N(tₓ) decreasing roughly as N^{-0.05} and the slope does not diverge; thus the transition is continuous, confirming that the product rule inherently fails to satisfy the SP. For the sum rule, model A still shows a decreasing G_N(tₓ) (SP violated), whereas models B and C display an almost constant G_N(tₓ) for large N and a slope that grows as N^{0.5}. Both criteria (α) and (β) are satisfied within the simulated size range, indicating a discontinuous EP when the SP is respected.

The paper also revisits the CDGM model (Cho–Da‑Goh–Mason). The original CDGM rule connects the two smallest clusters among four randomly chosen nodes, which ignores intra‑cluster edges and therefore violates the SP, leading to a continuous transition. By modifying the rule so that intra‑cluster edges are selected when present (models B and C), the authors obtain the same signatures of a discontinuous transition as in the sum‑rule B/C models.

In summary, the study demonstrates that the nature of the explosive percolation transition is not universal but depends critically on whether the edge‑selection dynamics faithfully implements the suppression principle. When the SP is fulfilled (as in the sum‑rule B and C, and the modified CDGM B/C), the giant component can emerge with a finite jump and a diverging slope, i.e., a discontinuous EP. When the SP is broken (product rule or any model that allows intra‑cluster edges to increase cluster size), the transition remains continuous. The authors caution that finite‑size effects and statistical errors at very large N prevent a definitive proof of discontinuity, but their extensive simulations strongly support the claim that satisfying the SP is essential for a truly discontinuous explosive percolation.