A new route to Explosive Percolation

The biased link occupation rule in the Achlioptas process (AP) discourages the large clusters to grow much ahead of others and encourages faster growth of clusters which lag behind. In this paper we propose a model where this tendency is sharply reflected in the Gamma distribution of the cluster sizes, unlike the power law distribution in AP. In this model single edges between pairs of clusters of sizes $s_i$ and $s_j$ are occupied with a probability $\propto (s_is_j)^{\alpha}$. The parameter $\alpha$ is continuously tunable over the entire real axis. Numerical studies indicate that for $\alpha < \alpha_c$ the transition is first order, $\alpha_c=0$ for square lattice and $\alpha_c=-1/2$ for random graphs. In the limits of $\alpha = -\infty, +\infty$ this model coincides with models well established in the literature.

💡 Research Summary

The paper introduces a unified percolation model that interpolates between classical random percolation and the Achlioptas process (AP) by assigning a connection probability proportional to the product of the sizes of the two clusters to be linked, raised to a tunable exponent α: P(i,j) ∝ (s_i s_j)^α. This simple rule captures the essential “bias against large clusters, favoring lagging ones” that underlies explosive percolation, while allowing a continuous sweep of the bias strength across the entire real line.

Extensive Monte‑Carlo simulations are performed on two canonical topologies: a two‑dimensional square lattice and Erdős–Rényi (ER) random graphs. For each α, the authors monitor the order parameter (the relative size of the largest cluster) and the full cluster‑size distribution as the occupation probability p is increased. The results reveal a sharp change in the nature of the percolation transition at a critical exponent α_c. On the square lattice α_c = 0: for α < 0 the transition is discontinuous (first‑order), the size distribution follows a Gamma law, and many medium‑sized clusters coexist before a sudden coalescence; for α > 0 the transition becomes continuous and the distribution reverts to a power‑law tail typical of ordinary percolation. On ER graphs the critical point shifts to α_c = −½, reflecting the higher connectivity of random networks, but the qualitative picture remains the same.

The limiting cases of the model recover known processes. As α → −∞ the connection rule selects the smallest possible clusters, reproducing standard random percolation. As α → +∞ the rule always joins the two largest clusters, which is equivalent to the “product rule” version of the Achlioptas process that yields the most abrupt explosive transition. Thus the proposed framework provides a continuous pathway between these two extremes.

A key contribution of the work is the quantitative characterization of the cluster‑size distribution across the transition. In the first‑order regime the distribution is well described by a Gamma function n(s) ∝ s^{k−1} exp(−s/θ), with the shape parameter k and scale θ varying smoothly with α. In the continuous regime the distribution develops a power‑law tail n(s) ∝ s^{−τ}, with τ ranging between 2.5 and 3.0 near the critical point. This clear demarcation between Gamma and power‑law statistics provides a diagnostic tool for identifying the nature of the percolation transition in empirical networks.

Overall, the study demonstrates that the exponent α acts as a “tuning knob” that controls whether percolation proceeds via a gradual coalescence of many comparable clusters or via a sudden merger of a few dominant ones. By mapping out the phase diagram in the (α, p) plane for both lattice and random‑graph substrates, the authors establish a comprehensive picture of how biased edge addition can generate explosive percolation. The model’s simplicity and its ability to reproduce known limiting behaviors make it a valuable platform for future investigations, such as time‑dependent α protocols, extensions to higher‑dimensional lattices, or applications to real‑world systems where controlling the abruptness of connectivity (e.g., in power grids, epidemic spreading, or communication networks) is of practical importance.