Explosive percolation on the Bethe lattice

Abstract

Based on the self-consistent equations of the order parameter P∞ and the mean cluster size S, we develop a novel self-consistent simulation (SCS) method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). 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]. Subsequent studies on variants of EP models on networks and lattices also argued to show the discontinuous transition [2-9]. In contrast Riordan and Warnke [13] analytically showed that the phase transition in AP model [1] on CG is continuous by use of the arbitrary connectivity of CG. Furthermore several studies also showed that the transition of variants of EP models on CG is continuous [10][11][12]. However, EP on CG was still reported to undergo discontinuous transition depending on the detail of cluster growth rule [14,15]. Therefore the transition nature of EP on CG is not still clear. Since the dimensionality of CG is infinite, physics on CG must satisfy the mean-field theory. In this sense the mean-field theory of explosive percolation is not still clearly understood.The Bethe lattice (infinite homogeneous Cayley tree) is physically a very important substrate or medium on which the mean-field theories for various physical models become exact [16]. The analytic treatments of magnetic models [17], percolation [16,18] and localization [16] on the Bethe lattice give important physical insights to the subsequent developments of the corresponding research fields. One of theoretical merits of the Bethe lattice is that one can setup the exact self-consistent equations (SCEs). In this letter, by use of the exact SCEs we develop a novel self-consistent simulation (SCS) method for arbitrary percolation process on the Bethe lattice. From SCS method, we precisely calculate the order parameter P ∞ and the average size S of finite clusters of the AP models with a product rule or a sum rule. The obtained P ∞ and S can clarify the transition nature of AP models * Electronic address: syook@khu.ac.kr † Corresponding author:ykim@khu.ac.kr in the infinite dimension exactly. Furthermore unlike AP on CG, the bond connections on the Bethe lattice are purely local. Since there have been some papers that EP models on lattices with local bond connections show the discontinuous transition [2,4,9], it is physically important to study EP models on the Bethe lattice or in the mean-field level with local connections. As we shall see, the transition of A...

Agglomerative percolation on the Bethe lattice and the triangular cactus

“…One of such indirect methods is a simulation method. We have developed a simulation method to solve self-consistent equations, which we call the self-consistent simulation [7]. In the self-consistent simulation, P mst (n, and S are estimated from Eqs.…”

“…In the self-consistent simulation, it should be careful to choose m ′ (< m) for a given m as addressed in Ref. [7]. If m ′ is too small, the clusters within the m ′ -generation tree cannot have physical properties of AP 2 enough to give physically plausible solutions for self-consistent equations (1) The results of the self-consistent simulation with z = 3, m = 20, m ′ = 9 for AP 2 are displayed in Figs.…”

“…[7]. In the method we originally used the fraction p of occupied bonds or sites, but we use the number n of clusters per site in this paper.…”

“…But subsequent studies on the explosive percolation have shown that the transition of the explosive percolation on CG or the mean-field transition is continuous [4][5][6][7].…”

“…Recently we have developed an exact self-consistent simulation method for an arbitrary percolation process on the Bethe lattice [7]. From the self-consistent simulation method, we have shown that the phase transition of the Achlioptastype explosive percolation [2] undergoes continuous transition regardless of the details of growth rules.…”