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

Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of α, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime αε[0.6,0.95] where it is known that only one giant component, denoted C(1) , initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime C(1) stops growing and eventually a second giant component, denoted C(2), emerges in a continuous percolation transition. The delay between the emergence of C(1) and C(2) and their asymptotic sizes both depend on the value of α and we establish by several techniques that there exists a bifurcation point α(c)=0.763±0.002. For αε[0.6,α(c)), C(1) stops growing the instant it emerges and the delay between the emergence of C(1) and C(2) decreases with increasing α. For αε(α(c),0.95], in contrast, C(1) continues growing into the supercritical regime and the delay between the emergence of C(1) and C(2) increases with increasing α. As we show, α(c) marks the minimal delay possible between the emergence of C(1) and C(2) (i.e., the smallest edge density for which C(2) can exist). We also establish many features of the continuous percolation of C(2) including scaling exponents and relations.

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“…There is much recent interest in understanding the formation mechanism of multiple giant components in percolation [38,40,54,55]. In Refs.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…In Ref. [54], the so called multi-ER model shows multiple giant components appear in a continuous transition. The formation of multiple giant components is attributed to the relatively low probability for merger of large components in the critical window.…”

Section: Summary and Discussionmentioning

confidence: 99%

Phase transitions in supercritical explosive percolation

et al. 2013

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“…[71] [13,69,70,[72][73][74][75] [ 76] Probability [53,77,78,80] [81] [81] [ 56,77,79,80,82] Hybrid [25,53,83] [ 25,83,84] DLCA [10,85] [85]…”

Section: Modelmentioning

confidence: 99%

“…regardless on whether or not they are occupied), while in Refs. [71,72] edges are only sampled from the set of unoccupied bonds. In Refs.…”

Section: The Bfw Modelmentioning

confidence: 99%

Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization

et al. 2016

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Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in complex networks' structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the network's structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks' dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and debates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.

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“…In the thermodynamic limit, at the percolation threshold, the rate of change of the order parameter diverges [51,52]. Therefore, we consider as one estimator the average fraction of occupied bonds or sites p c,J at which the largest change in P ∞ occurs [36,[53][54][55][56][57][58][59][60][61]. The second moment of the cluster size distribution is defined as,…”

Section: Determining the Percolation Thresholdmentioning

confidence: 99%

Stacked triangular lattice: Percolation properties

2013

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The stacked triangular lattice has the shape of a triangular prism. In spite of being considered frequently in solid state physics and materials science, its percolation properties have received few attention. We investigate several non-universal percolation properties on this lattice using Monte Carlo simulation. We show that the percolation threshold is $p_c^\text{bond}=0.186\;02\pm0.000\;02$ for bonds and $p_c^\text{site}=0.262\;40\pm0.000\;05$ for sites. The number of clusters at the threshold per site is $n_c^\text{bond}=0.284\;58\pm0.000\;05$ and $n_c^\text{site}=0.039\;98\pm0.000\;05$. The stacked triangular lattice is a convenient choice to study the RGB model [Sci. Rep. {\bf 2}, 751 (2012)]. We present results on this model and its scaling behavior at the percolation threshold

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

Cited by 12 publications

References 39 publications

Phase transitions in supercritical explosive percolation

Phase transitions in supercritical explosive percolation

Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization

Stacked triangular lattice: Percolation properties

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