Explosive percolation: Unusual transitions of a simple model

Bastas, Nikolaos; Giazitzidis, Paraskevas; Maragakis, Michael; Kosmidis, Kosmas

doi:10.1016/j.physa.2014.03.085

Cited by 36 publications

(34 citation statements)

References 66 publications

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“…In turn, our result means that adding invisible states turns transition in the bond-percolation model into strong first-order. It is also worth mentioning here other mechanisms that are known to sharpen percolation transitions, such as those delivering explosive [38,39] or bootstrap [40] percolation.…”

Section: Discussionmentioning

confidence: 99%

Marginal dimensions of the Potts model with invisible states

Krasnytska

Sarkanych

Berche

et al. 2016

J. Phys. A: Math. Theor.

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Abstract.We reconsider the mean-field Potts model with q interacting and r non-interacting (invisible) states. The model was recently introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where the Z q -symmetry is spontaneously broken. We analyse the marginal dimensions of the model, i.e., the value of r at which the order of the phase transition changes. In the q = 2 case, we determine that value to be r c = 3.65(5); there is a second-order phase transition there when r < r c and a first-order one at r > r c . We also analyse the region 1 ≤ q < 2 and show that the change from second to first order there is manifest through a new mechanism involving two marginal values of r. The q = 1 limit gives bond percolation and some intermediary values also have known physical realisations. Above the lower value r c1 , the order parameters exhibit discontinuities at temperaturet below a critical value t c . But, provided r > r c1 is small enough, this discontinuity does not appear at the phase transition, which is continuous and takes place at t c . The larger value r c2 marks the point at which the phase transition at t c changes from second to first order. Thus, for r c1 < r < r c2 , the transition at t c remains second order while the order parameter has a discontinuity att. As r increases further,t increases, bringing the discontinuity closer to t c . Finally, when r exceeds r c2 t coincides with t c and the phase transition becomes first order. This new mechanism indicates how the discontinuity characteristic of first order phase transitions emerges.

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Section: Discussionmentioning

confidence: 99%

Marginal dimensions of the Potts model with invisible states

Krasnytska

Sarkanych

Berche

et al. 2016

J. Phys. A: Math. Theor.

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“…Furthermore, it is natural to expect that close to p c nearly equal sized clusters, waiting to merge, are so great in number that occupation of a few bonds results in an abrupt global connection and thus the name "Explosive Percolation" (EP). Through their seminal paper Achlioptas et al claimed for the first time that EP can describe the first order phase transition (see for recent reviews in [7,8]). Their results jolted the scientific community through a series of claims, unclaims and counter-claims [9][10][11][12][13][14][15][16][17][18] .…”

Section: Introductionmentioning

confidence: 99%

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

Rahman

Hassan

2017

Phys. Rev. E

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In this article, we investigate explosive bond percolation (EBP) with product rule, formally known as Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice (WPSL). One of the key features of the EBP transition is the delay, compared to corresponding random bond percolation (RBP), in the onset of spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, albeit ultimately proved wrong, that explosive percolation (EP) exhibits first order transition. In the case of EP, much efforts were devoted to resolving the issue of its order of transition and almost no effort being devoted to find critical point, critical exponents etc., to classify it into universality classes. This is in sharp contrast to the classical random percolation. We do not even know all the exponents of EP for regular planar lattice or for Erdös-Renyi network. We first find numerically the critical point pc and then obtain all the critical exponents β, γ, ν as well as the Fisher exponent τ and the fractal dimension d f of the spanning cluster. We also compare our results for EBP with those of the RBP and find that all the exponents of EBP obeys the same scaling relations as do the RBP. Our findings suggests that EBP is no special except the fact that the exponent β is unusually small compared to that of RBP.

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“…To better quantify the abruptness of the DPR transition, we measure the size of the largest jump in the order parameter ∆C max N during each realization, then average over many realizations. This type of convergence criteria is common among explosive percolation studies [32][33][34][35] as it gives insight into how the transition behaves in the thermodynamic limit and indicates whether the transition is first-order or second-order. For increasing system size, the largest jump will decay as a power law when the transition is second-order, ∆C max N ∼ N −ω , whereas if there is a discontinuity that survives in the thermodynamic limit then ∆C max N will asymptote to a constant value, signaling that the transition is first-order.…”

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confidence: 99%

Degree product rule tempers explosive percolation in the absence of global information

2018

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We introduce a guided network growth model, which we call the degree product rule process, that uses solely local information when adding new edges. For small numbers of candidate edges our process gives rise to a second-order phase transition, but becomes first-order in the limit of global choice. We provide the set of critical exponents required to characterize the nature of this percolation transition. Such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes.Introduction.-Network based approaches continue to see growing applications in a wide array of fields, from epidemiology [1, 2] to finance [3,4], neuroscience [5,6], and machine learning [7]. As we increasingly rely on networks, understanding how they form out of complex conditions becomes all the more consequential [8][9][10][11][12]. Many the networks we entrust to support our modernized society-transportation, financial, social, etc.-are formed with some amount of agency, meaning that potential new members have control over how they connect and interact with the network. This agency can lead to markedly different behavior compared to the classical case of purely random network growth [13]. In particular, networks subject to competitive edge addition break time-reversal symmetry, as there is no well-defined method for running the process in reverse that achieves a statistically identical growth curve [14]. Furthermore, edge competition can be used as a means of control over cluster growth and connectivity within a growing network. Depending on the desired outcome (delayed connectivity for contagion spreading, increased connectivity for communication networks, etc.), intervening on growing networks can help produce more specialized and responsive networks.

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Explosive percolation: Unusual transitions of a simple model

Cited by 36 publications

References 66 publications

Marginal dimensions of the Potts model with invisible states

Marginal dimensions of the Potts model with invisible states

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

Degree product rule tempers explosive percolation in the absence of global information

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