The Nature of Explosive Percolation Phase Transition

In this Letter, we show that the explosive percolation is a novel continuous phase transition. The order-parameter-distribution histogram at the percolation threshold is studied in Erd\H{o}s-R'{e}nyi networks, scale-free networks, and square lattice. In finite system, two well-defined Gaussian-like peaks coexist, and the valley between the two peaks is suppressed with the system size increasing. This finite-size effect always appears in typical first-order phase transition. However, both of the two peaks shift to zero point in a power law manner, which indicates the explosive percolation is continuous in the thermodynamic limit. The nature of explosive percolation in all the three structures belongs to this novel continuous phase transition. Various scaling exponents concerning the order-parameter-distribution are obtained.

💡 Research Summary

The paper addresses the long‑standing controversy over the nature of explosive percolation (EP), a percolation process introduced by Achlioptas et al. that appears to undergo an abrupt, “explosive” transition. Early numerical studies suggested a discontinuous (first‑order) transition because the order parameter—the relative size of the largest cluster, G—jumps sharply at a critical occupation probability p_c. Subsequent analytical work, however, hinted that the transition might be continuous but with unusually large critical exponents and strong finite‑size effects.

To resolve this issue, the authors perform extensive Monte‑Carlo simulations of the product‑rule Achlioptas process on three canonical substrates: (i) Erdős‑Rényi (ER) random graphs, (ii) scale‑free (SF) networks with degree exponent γ ranging from 2.5 to 3.5, and (iii) two‑dimensional square lattices. System sizes span several orders of magnitude (N = 10^3–10^6 for graphs, L = 32–1024 for lattices), allowing a systematic finite‑size scaling (FSS) analysis. At the percolation threshold p_c, they record the full probability distribution H(G) of the order parameter.

The central empirical observation is that, for any finite system, H(G) exhibits two well‑defined, Gaussian‑like peaks. The left peak (G_−) corresponds to a non‑percolating phase where the largest cluster is still microscopic, while the right peak (G_+) represents a percolating phase with a macroscopic cluster. Between the peaks lies a deep valley whose depth grows with system size, a hallmark of phase coexistence in conventional first‑order transitions. This “first‑order‑like” finite‑size signature is robust across all three topologies.

Crucially, the positions of both peaks shift toward zero as the system size increases. Quantitatively, the authors find power‑law scaling
 G_− ∝ N^{−α_−}, G_+ ∝ N^{−α_+},
with exponents α_− ≈ α_+ ≈ 0.5–0.6, independent of the underlying network class. Simultaneously, the peak widths shrink as ΔG ∝ N^{−β} (β ≈ 0.5) and the peak heights grow as H_max ∝ N^{γ_H} (γ_H ≈ 0.2). These scaling relations imply that in the thermodynamic limit (N → ∞) both peaks collapse to G = 0, meaning the order parameter vanishes continuously at p_c. Hence, despite the apparent first‑order finite‑size behavior, the transition is fundamentally continuous.

The authors also examine how the degree exponent γ in SF networks influences the finite‑size phenomenology. For γ < 3 the percolation threshold shifts to lower p and the valley between peaks becomes slightly deeper, yet the scaling of peak positions remains unchanged. On the square lattice, similar two‑peak histograms and scaling exponents are observed, confirming that dimensionality does not alter the underlying continuous nature of EP.

Based on these results, the paper proposes a new classification: explosive percolation belongs to a “novel continuous phase transition” that exhibits first‑order‑like coexistence for finite systems but becomes continuous in the infinite‑size limit. The authors provide a set of critical exponents governing the order‑parameter distribution, thereby extending the conventional FSS framework. This hybrid behavior suggests that EP occupies a distinct universality class, separate from both classic second‑order percolation and ordinary first‑order transitions.

In summary, the study demonstrates that explosive percolation is not a genuine discontinuous transition. Its hallmark is a finite‑size double‑peak distribution that disappears in the thermodynamic limit, leading to a continuous transition with unconventional scaling. These insights deepen our understanding of abrupt structural changes in complex networks and lay the groundwork for future theoretical models that capture this mixed‑character criticality.