Percolation Transitions in Scale-Free Networks under the Achlioptas Process

Cho, Y. S.; Kim, J. S.; Park, Juyong; Kahng, B.; Kim, D.

doi:10.1103/physrevlett.103.135702

Cited by 175 publications

(198 citation statements)

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“…[1] to describe a sudden appearance of a macroscopic cluster in a network growth model with the so-called product rule considered on the complete graph. This growth rule, named as the Achlioptas process (AP), is then studied on the two-dimensional lattice [2,3] and on the scalefree networks [4][5][6] as well, yielding similar results. That suddenness has been widely believed to indicate a discontinuity at the percolation transition in the thermodynamic limit [7,8], and the similar explosiveness has been observed with the other growth rules proposed later [9][10][11][12][13].…”

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confidence: 79%

“…Sublinearity is estimated only by 6%, which may call for a huge system size like N ∼ 10 17 (beyond the present computing capability) to reach a reasonable scaling regime (g(t u ) 0.1) and get any sensible extrapolation to the thermodynamic limit. Most of previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13]15] basically depend on the data in this supercritical regime (t > t c ). Nevertheless, the scaling plot with this natural cutoff shows a reasonable collapse including the dip structure at the end, but involving big statistical errors (not shown here).…”

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confidence: 99%

“…It may be natural to use the information above the transition point (t > t c ) in order to prove the (non-)existence of the gap or estimate the gap size. Thus, most of previous studies have focussed on this information [1][2][3][4][5][6][7][8][9][10][11][12][13]15], but could not provide a definitive answer due to the extremely slow convergence of the order parameter in system size. In this work, we took the opposite approach.…”

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confidence: 99%

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Continuity of the explosive percolation transition

2011

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The explosive percolation problem on the complete graph is investigated via extensive numerical simulations. We obtain the cluster-size distribution at the moment when the cluster size heterogeneity becomes maximum. The distribution is found to be well described by the power-law form with the decay exponent τ = 2.06(2), followed by a hump. We then use the finite-size scaling method to make all the distributions at various system sizes up to N = 2 37 collapse perfectly onto a scaling curve characterized solely by the single exponent τ . We also observe that the instant of that collapse converges to a well-defined percolation threshold from below as N → ∞. Based on these observations, we show that the explosive percolation transition in the model should be continuous, contrary to the widely-spread belief of its discontinuity.PACS numbers: 64.60.ah, 64.60.aq, 36.40.Ei The term explosive percolation was proposed in Ref.[1] to describe a sudden appearance of a macroscopic cluster in a network growth model with the so-called product rule considered on the complete graph. This growth rule, named as the Achlioptas process (AP), is then studied on the two-dimensional lattice [2,3] and on the scalefree networks [4][5][6] as well, yielding similar results. That suddenness has been widely believed to indicate a discontinuity at the percolation transition in the thermodynamic limit [7,8], and the similar explosiveness has been observed with the other growth rules proposed later [9][10][11][12][13]. These observations of the explosiveness have drawn much interest due to the striking difference from the wellknown continuous transition in the standard percolation models [14]. However, in our point of view, the explosiveness has not been carefully investigated as yet enough to draw a decisive conclusion on the discontinuity, and possibly just represents an extremely steep but still continuous transition.Friedman and Landsberg [9] have suggested the argument of the powder keg as a circumstantial description to explain the apparent discontinuity of the explosive percolation transition. Meanwhile, da Costa et al. [15] have reported that the explosive percolation is actually continuous for a modified version of the AP by analytically deriving the critical scaling relations based on numerical observations of power-law critical distribution of cluster size [16]. In this Letter, we try to unmask the (dis)continuity in a systematic and direct way by performing a careful finite-size-scaling analysis at newly introduced pseudo-transition points for finite systems and show that the explosive percolation transition on the complete graph is indeed continuous in the thermodynamic limit.The model we study is the AP with the product rule on the complete graph [1]. Start with N nodes with all links unoccupied. At each step, choose two possible unoccupied links randomly between nodes. Then, select the link merging two clusters with a smaller product of the two cluster sizes. Here, a cluster is defined as a set of nodes connected each other via...

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confidence: 99%

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Continuity of the explosive percolation transition

2011

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“…A mean-field theory is developed to analytically reveal the property of this phase transition. Recently, explosive or discontinuous transitions in complex networks have received growing attention since the discovery of an abrupt percolation transition in random networks [8,9] and scale-free networks [10,11]. Later studies affirmed that this transition is actually continuous but with an unusual finite size scaling [12][13][14], yet many related models show truly discontinuous and anomalous transitions (cf.…”

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confidence: 99%

“…The other possible solutions can be obtained by numerically iterating Eq. (11). Oncem is found, we can immediately calculate m k by Eq.…”

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First-order phase transition in a majority-vote model with inertia

et al. 2017

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We generalize the original majority-vote model by incorporating an inertia into the microscopic dynamics of the spin flipping, where the spin-flip probability of any individual depends not only on the states of its neighbors, but also on its own state. Surprisingly, the order-disorder phase transition is changed from a usual continuous type to a discontinuous or an explosive one when the inertia is above an appropriate level. A central feature of such an explosive transition is a strong hysteresis behavior as noise intensity goes forward and backward. Within the hysteresis region, a disordered phase and two symmetric ordered phases are coexisting and transition rates between these phases are numerically calculated by a rare-event sampling method. A mean-field theory is developed to analytically reveal the property of this phase transition. Recently, explosive or discontinuous transitions in complex networks have received growing attention since the discovery of an abrupt percolation transition in random networks [8,9] and scale-free networks [10,11]. Later studies affirmed that this transition is actually continuous but with an unusual finite size scaling [12][13][14], yet many related models show truly discontinuous and anomalous transitions (cf.[15] for a recent review). Striking different from continuous phase transitions, in an explosive transition an infinitesimal increase of the control parameter can give rise to a considerable macroscopic effect. Subsequently, an explosive phenomenon was found in the dynamics of cascading failures in interdependent networks [16][17][18], in contrast to the secondorder continuous phase transition found in isolated networks. More recently, such explosive phase transitions have been reported in various systems, such as explosive synchronization due to a positive correlation between the degrees of nodes and the natural frequencies of the oscillators [19][20][21] [25][26][27].In this paper we report an explosive order-disorder phase transition in a generalized majority-vote (MV) model by incorporating the effect of individuals' inertia (called inertial MV model ). The MV model is one of the simplest nonequilibrium generalizations of the Ising model that displays a continuous order-disorder phase transition at a critical value of noise [28]. It has been extensively studied in the context of complex networks, including random graphs [29,30], small world networks [31][32][33], and scale-free networks [34,35]. However, the continuous nature of the order-disorder phase transition is not affected by the topology of the underlying networks [36]. In our model, we have included a substantial change to make it more realistic, namely the state update of each node depends not only on the states of its neighboring nodes, but also on its own state. In fact, in a social or biological context individuals have a tendency for beliefs to endure once formed. In a recent experimental study, behavioral inertia was found to be essential for collective turning of starling flocks [37]. We refer this m...

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Explosive Percolation Processes

D’Souza¹

2021

Complex Media and Percolation Theory

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Percolation Transitions in Scale-Free Networks under the Achlioptas Process

Cited by 175 publications

References 9 publications

Continuity of the explosive percolation transition

Continuity of the explosive percolation transition

First-order phase transition in a majority-vote model with inertia

Explosive Percolation Processes

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