Explosive site percolation with a product rule

Abstract

We study the site percolation under Achlioptas process (AP) with a product rule in a 2 − dimensional (2D) square lattice. From the measurement of the cluster size distribution, Ps, we find that Ps has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the Ps distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in 2D. Moreover we also show that the transition nature from the product rule is not the same as that from a sum rule in 2D. PACS numbers: 64.60.ah, 64.60.De, 05.70.Fh, 64.60.Bd The percolation transition describing the emergence of large-scale connectivity in lattice systems or complex networks has been extensively studied in statistical mechanics and related fields due to its possible applications to various phenomena such as sol-gel transition and polymerization, resistor networks, and epidemic spreading [1]. When the occupation probability of node (site) is lower than certain threshold p c , all the clusters are microscopic. As the occupation probability increases, the macroscopically connected cluster emerges. Such transition in the ordinary percolation is a continuous transition [1].On the other hand, there have been several attempts to find a percolation model which undergoes a discontinuous transition. The discontinuous percolation transition can be found in the modelling of magnetic systems with significant competition between exchange and crystal-field interactions [2,3]. The similar phenomena has been found in financial systems [4], in which two equally probable phases exit. Other examples of the discontinuous transition in percolation are the formation of infinite cluster under a central-force [5] and the cascade of failure in interdependent networks [6].Recently, Achlioptas et al.[7] suggested a simple process in which the growth of large clusters is systematically suppressed and the process is usually called as Achlioptas process (AP). Based on the analysis of transition interval it was argued that the percolation transition under AP is explosive and discontinuous. Several variant of models have been investigated to understand the general properties and conditions which cause such non-trivial discontinuous transition [8][9][10][11]. Some examples of such non-trivial transition has been found in nano-tube based system [11], protein homology network [12], and community formation [13].However, more recent studies on the percolation transition under AP reveals several evidences which strongly suggest that the transition can be continuous. For example, da Costa et al. [14] argued that the transition in the complete graph (CP) is continuous, even though the order parameter exponent is very small (β ≃ 0.056). From the measurement of the cluster size distribution Lee et al.[15] also argued that the transition in CP is continuous. Grassberger et al.[16] also argued that the transition, even in the low-dimension...

Bond-site duality and nature of the explosive-percolation phase transition on a two-dimensional lattice

“…Even though there are some studies on the AP in two dimensions (2D) [3,7,[9][10][11][12], the transition nature in lower dimensions is still not fully understood. For example, the bond percolation under the AP with a product rule was first argued to show a discontinuous transition [3,9].…”

“…However, based on the measurement of the largest cluster distribution, Grassberger et al [7] argued that the bond percolation with the same product rule in 2D undergoes continuous transition [7]. The site percolation under the AP with a product rule in 2D has been proved to undergo a discontinuous transition based on a detailed analysis of cluster size distribution and hysteresis [11]. In contrast, Bastas et al argued that the site percolation under the AP with a sum rule in 2D undergoes continuous transition based on the finite-size scaling analysis with relatively small system sizes [12].…”

Explosive percolation on the Bethe lattice

“…By applying SCS to the well-known percolation models, random bond percolation and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing the key functions obtained from SCSs for the Achlioptas processes (APs) with a product rule and a sum rule to the prototype functions, we show that the percolation transition of AP models on the Bethe lattice is continuous regardless of details of growth rules.-- [2][3][4][5][6][7][8][9][10][11][12]. Achlioptas process (AP) was originally argued to show the discontinuous phase transition on the complete graph (CG) by suppressing growth of large clusters [1].…”

“…-- [2][3][4][5][6][7][8][9][10][11][12]. Achlioptas process (AP) was originally argued to show the discontinuous phase transition on the complete graph (CG) by suppressing growth of large clusters [1].…”

Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

“…The EP model was first implemented on the ER network. The idea was then extended to other planar lattices and to scale-free networks [9,24]. As many variants of the EP model were introduced, it became more apparent that EP actually describes continuous phase transition.…”