Critical behavior of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-core percolation: Numerical studies

Lee, Deokjae; Jo, Minjae; Kahng, B.

doi:10.1103/physreve.94.062307

Cited by 29 publications

(35 citation statements)

References 25 publications

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“…This is in line with our analysis of the limit where C → ∞(N ) [2]. There we showed that, rather than recovering standard percolation in this case, we instead recover k-core percolation with k = 2 [10,11,36]. On the other hand, color connectivity can be expressed in terms of variables v Q using Eq.…”

Section: A Critical Behaviorsupporting

confidence: 87%

Color-avoiding percolation

2017

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Many real world networks have groups of similar nodes which are vulnerable to the same failure or adversary. Nodes can be colored in such a way that colors encode the shared vulnerabilities. Using multiple paths to avoid these vulnerabilities can greatly improve network robustness, if such paths exist. Color-avoiding percolation provides a theoretical framework for analyzing this scenario, focusing on the maximal set of nodes which can be connected via multiple color-avoiding paths. In this paper we extend the basic theory of color-avoiding percolation that was published in S. M. Krause et al. [Phys. Rev. X 6, 041022 (2016)]2160-330810.1103/PhysRevX.6.041022. We explicitly account for the fact that the same particular link can be part of different paths avoiding different colors. This fact was previously accounted for with a heuristic approximation. Here we propose a better method for solving this problem which is substantially more accurate for many avoided colors. Further, we formulate our method with differentiated node functions, either as senders and receivers, or as transmitters. In both functions, nodes can be explicitly trusted or avoided. With only one avoided color we obtain standard percolation. Avoiding additional colors one by one, we can understand the critical behavior of color-avoiding percolation. For unequal color frequencies, we find that the colors with the largest frequencies control the critical threshold and exponent. Colors of small frequencies have only a minor influence on color-avoiding connectivity, thus allowing for approximations.

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Section: A Critical Behaviorsupporting

confidence: 87%

Color-avoiding percolation

2017

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“…The values of γ/ν for different values of g are also verified by estimating the average cluster size of all the finite clusters (excluding the spanning cluster) at their respective percolation thresholds. However, there are evidences in some other percolation models such as k-core percolation [36] that the scaling behavior of order parameter fluctuation and that of the average cluster size are not identical.…”

Section: (B)mentioning

confidence: 99%

Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition

Roy¹,

Santra²

2018

J. Stat. Mech.

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A random growth lattice filling model of percolation with touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty sites and the clusters are grown from these nucleation centers with a tunable growth probability g. As the growth probability g is varied from 0 to 1 two distinct regimes are found to occur. For g ≤ 0.5, the model exhibits continuous percolation transitions as ordinary percolation whereas for g ≥ 0.8 the model exhibits discontinuous percolation transitions. The discontinuous transition is characterized by discontinuous jump in the order parameter, compact spanning cluster and absence of power law scaling of cluster size distribution. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously till the crossover from continuous to discontinuous transition is completed.

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“…A hybrid phase transition is a discontinuous transition but with criticality. Such a transition occurs in several a) Electronic mail: bkahng@snu.ac.kr complex systems, including cascade failure of interdependent networks 5 , k-core [6][7][8] , cooperative epidemic spreading [9][10][11][12] and synchronization [13][14][15][16][17] . However, a possible hysteresis together with critical behavior in hybrid phase transitions has been rarely reported thus far, even though hysteresis in discontinuous transitions was investigated in explosive synchronization [18][19][20] and percolation [21][22][23][24][25][26][27][28][29] in complex systems.…”

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confidence: 99%

“…For percolation, hybrid percolation transitions (HPTs) may be classified into two categories: one occurring during the cascade process as in k-core percolation [5][6][7][8][9][10][11][12] , and the other during the cluster merging process [30][31][32] . For the former case, dynamics proceed only in one direction of decreasing occupation probability, and a reverse process may not be well defined, i.e., a cascade cluster aggregation process is hardly imagined.…”

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confidence: 99%

Hysteresis and criticality in hybrid percolation transitions

Park

Kahng

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Phase transitions (PTs) are generally classified into second-order and first-order transitions, each exhibiting different intrinsic properties. For instance, a first-order transition exhibits latent heat and hysteresis when a control parameter is increased and then decreased across a transition point, whereas a second-order transition does not. Recently, hybrid percolation transitions (HPTs) are issued in diverse complex systems, in which the features of first-order and second-order PTs occur at the same transition point. Thus, the question whether hysteresis appears in an HPT arises. Herein, we investigate this fundamental question with a socalled restricted Erdős-Rényi random network model, in which a cluster fragmentation process is additionally proposed. The hysteresis curve of the order parameter was obtained. Depending on when the reverse process is initiated, the shapes of hysteresis curves change, and the critical behavior of the HPT is conserved throughout the forward and reverse processes.

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Critical behavior ofk-core percolation: Numerical studies

Cited by 29 publications

References 25 publications

Color-avoiding percolation

Color-avoiding percolation

Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition

Hysteresis and criticality in hybrid percolation transitions

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