Gaussian model of explosive percolation in three and higher dimensions

Schrenk, K. J.; Araújo, Nuno A. M.; Herrmann, Hans J.

doi:10.1103/physreve.84.041136

Cited by 40 publications

(62 citation statements)

References 60 publications

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“…It is the local optimization rule that leads to continuity of the explosive percolation transition in these models. In more exotic models employing various global optimization algorithms and their variations, discontinuities may occur [28][29][30][31][32][33][34].…”

Section: Discussionmentioning

confidence: 99%

Critical exponents of the explosive percolation transition

et al. 2014

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In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptaslike algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition". We have shown that this transition is actually continuous (second-order) though with an anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second order transition for a representative set of explosive percolation models with different number of choices. The method is based on gluing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.

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Section: Discussionmentioning

confidence: 99%

Critical exponents of the explosive percolation transition

et al. 2014

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“…Besides the Achlioptas process, many new models such as cluster aggregation model [15][16][17], Gaussian model [18,19], Hamiltonian approach [20] have been devised and studied, showing that discontinuous transition can be made when the evolution mechanism produces many clusters which are relatively large in the subcritical regime [21]. Although some numerical [22][23][24][25] and theoretical results [26] demonstrated that explosive percolation in the original Achlioptas processes is actually continuous in the mean-field limit, discontinuous transition indeed exists in other alternative models, e.g., the global competitive percolation process [27,28].…”

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Formation mechanism and size features of multiple giant clusters in generic percolation processes

Zhang

Wei²,

Guo³

et al. 2012

Phys. Rev. E

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Percolation is one of the most widely studied models in which a unique giant cluster emerges after the phase transition. Recently, a new phenomenon that multiple giant clusters are observed in so called BFW model has attracted much attention, and how multiple giant clusters could emerge in generic percolation processes on random networks will be concerned in this paper. By introducing the merging probability and inspecting the distinct mechanisms which contribute to the growth of largest clusters, a sufficient condition to generate multiple stable giant clusters is given. Based on the above results, the BFW model and a new multi-ER model given by us are analyzed, and the mechanism of multiple giant clusters of these two models is revealed. Furthermore, large fluctuations are observed in the size of multiple giant clusters in many models, but the sum size of all giant clusters exhibits self-averaging as that in the size of unique giant cluster in ordinary percolation. Besides, the growth modes of different giant clusters are discussed, and we find that the large fluctuations observed are mainly due to the stochastic behavior of the evolution in the critical window. For all the discussion above, numerical simulations on BFW model and multi-ER model are done, which strongly support our analysis. The investigation of merging probability and the growth mechanisms of largest clusters provides insight for the essence of multiple giant clusters in the percolation processes and can be instructive for modeling or analyzing real-world networks consisting of many large clusters.

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“…Among others [14][15][16][17][18][19], a model called spanning cluster avoiding (SCA) was introduced [20] aiming to generate a DPT. The DPT of the SCA model is rather trivial, for the percolation threshold is placed at p c = 1 in the thermodynamic limit, but for finite-sized systems p c < 1.…”

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confidence: 99%

Crossover behavior of conductivity in a discontinuous percolation model

et al. 2014

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When conducting bonds are occupied randomly in a two-dimensional square lattice, the conductivity of the system increases continuously as the density of those conducting bonds exceeds the percolation threshold. Such a behavior is well known in percolation theory; however, the conductivity behavior has not been studied yet when the percolation transition is discontinuous. Here we investigate the conductivity behavior through a discontinuous percolation model evolving under a suppressive external bias. Using effective medium theory, we analytically calculate the conductivity behavior as a function of the density of conducting bonds. The conductivity function exhibits a crossover behavior from a drastically to a smoothly increasing function beyond the percolation threshold in the thermodynamic limit. The analytic expression fits well our simulation data. The concept of percolation transition has played a central role as a model for the formation of a spanning cluster connecting two opposite edges of a system in Euclidean space as a control parameter p is increased beyond a certain threshold p c [1]. This model has been used to study many phenomena such as metal-insulator transitions and sol-gel transitions. The order parameter P ∞ of percolation transition is defined as the probability that a bond belongs to a spanning cluster, which increases in the form, where p is a control parameter indicating the fraction of occupied bonds and β is the critical exponent related to the order parameter. As an application of percolation model, one can construct a random resistor network in which each occupied bond is regarded as a resistor with unit resistance, and the system is in contact with two bus bars at the opposite edges of the system. When a voltage difference is applied between these two bus bars, the system is in a insulating state for p < p c , but changes to conducting state for p > p c , due to the formation of several conducting paths at p c . Above p c , the conductivity increases continuously as g ∼ (p − p c ) µ , where µ is the conductivity exponent [2].Recently the subject of discontinuous percolation transition (DPT) has been a central issue [3][4][5][6][7][8][9][10][11][12] with, for example, applicability to cascading failures in complex networks [13]. Among others [14][15][16][17][18][19], a model called spanning cluster avoiding (SCA) was introduced [20] aiming to generate a DPT. The DPT of the SCA model is rather trivial, for the percolation threshold is placed at p c = 1 in the thermodynamic limit, but for finite-sized systems p c < 1. Here, we study the conductivity as a function of p in finite-sized systems for the SCA model. Indeed, we find that the conductivity increases drastically just after the percolation threshold and then exhibits a crossover to a smoothly increasing behavior. Such crossover has never been reported, though it is meaningful, as, a drastic change of conductivity in random resistor networks can find application, for example, on resistance switching phenomena in non-volatile memory...

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Gaussian model of explosive percolation in three and higher dimensions

Cited by 40 publications

References 60 publications

Critical exponents of the explosive percolation transition

Critical exponents of the explosive percolation transition

Formation mechanism and size features of multiple giant clusters in generic percolation processes

Crossover behavior of conductivity in a discontinuous percolation model

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