Recent advances and open challenges in percolation

Araújo, Nuno A. M.; Grassberger, Peter; Kahng, B.; Schrenk, K. J.; Ziff, Robert M.

doi:10.1140/epjst/e2014-02266-y

Cited by 132 publications

(115 citation statements)

References 190 publications

(232 reference statements)

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“…To verify this we show first (in Fig. 22) the order parameter S (the density of the largest viable cluster) as a function of p. We see that the curves for different sizes N intersect indeed in a single point, with p c = 0.22614(1) and S c = 0.31 (1). This jump of S is a clear sign of a discontinuous transition, although the rise of S for p > p c indicates that the transition is also, as for ER networks, hybrid.…”

Section: -Dimensional Latticesmentioning

confidence: 70%

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth

Grassberger

2015

Phys. Rev. E

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The "SOS" in the title does not refer to the international distress signal, but to "solid-on-solid" (SOS) surface growth. The catastrophic cascades are those observed by Buldyrev et al. in interdependent networks, which we re-interpret as multiplex networks with agents that can only survive if they mutually support each other, and whose survival struggle we map onto an SOS type growth model. This mapping not only reveals non-trivial structures in the phase space of the model, but also leads to a new and extremely efficient simulation algorithm. We use this algorithm to study interdependent agents on duplex Erdös-Rényi (ER) networks and on lattices with dimensions 2, 3, 4, and 5. We obtain new and surprising results in all these cases, and we correct statements in the literature for ER networks and for 2-d lattices. In particular, we find that d = 4 is the upper critical dimension, that the percolation transition is continuous for d ≤ 4 but -at least for d = 3 -not in the universality class of ordinary percolation. For ER networks we verify that the cluster statistics is exactly described by mean field theory, but find evidence that the cascade process is not. For d = 5 we find a first order transition as for ER networks, but we find also that small clusters have a nontrivial mass distribution that scales at the transition point. Finally, for d = 2 with intermediate range dependency links we propose a scenario different from that proposed in W. Li et al., PRL 108, 228702 (2012).

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Section: -Dimensional Latticesmentioning

confidence: 70%

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth

Grassberger

2015

Phys. Rev. E

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“…Nonequilibrium dynamic transitions driven by cascade dynamics on complex networks have attracted considerable attention recently [1][2][3]. The spreading of epidemic disease on complex networks [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] is an instance, in which a pathogen is transmitted from an infected node (e.g., a person) to a susceptible neighbor, who then becomes infected with a certain probability.…”

Section: Introductionmentioning

confidence: 99%

Critical behavior of a two-step contagion model with multiple seeds

2017

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A two-step contagion model with a single seed serves as a cornerstone for understanding the critical behaviors and underlying mechanism of discontinuous percolation transitions induced by cascade dynamics. When the contagion spreads from a single seed, a cluster of infected and recovered nodes grows without any cluster merging process. However, when the contagion starts from multiple seeds of O(N ) where N is the system size, a node weakened by a seed can be infected more easily when it is in contact with another node infected by a different pathogen seed. This contagion process can be viewed as a cluster merging process in a percolation model. Here, we show analytically and numerically that when the density of infectious seeds is relatively small but O(1), the epidemic transition is hybrid, exhibiting both continuous and discontinuous behavior, whereas when it is sufficiently large and reaches a critical point, the transition becomes continuous. We determine the full set of critical exponents describing the hybrid and the continuous transitions. Their critical behaviors differ from those in the single-seed case.

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“…This percolation theory has been used for understanding percolation-related diverse phenomena such as conductor-insulator transitions [3], the resilience of systems [4][5][6], the formation of public opinion [7,8], and the spread of disease in a population [9,10]. The percolation transition is known to be one of the most robust continuous transitions [1,11].Recently, however, many abrupt percolation transitions have been observed in complex systems [12][13][14][15][16][17][18], for instance, large-scale blackouts in power grid systems [19] and pandemics [20], in which the order parameter increases abruptly at a transition point. Among those transitions, an HPT has attracted substantial attention.…”

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confidence: 99%

Critical phenomena of a hybrid phase transition in cluster merging dynamics

et al. 2017

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Recently, a hybrid percolation transitions (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. In spite of considerable effort to develop the theory of HPT, it is still incomplete, particularly when the transition is induced by cluster merging dynamics. Here, we aim to develop a theoretical framework of the HPT induced by such dynamics. We find that two correlation-length exponents are necessary for characterizing the giant cluster and finite clusters, respectively. Finite-size scaling method for the HPT is also introduced. The conventional formula of the fractal dimension in terms of the critical exponents is not valid. Neither the giant nor finite clusters are fractals but they have fractal boundaries.Percolation has long served as a simple model that undergoes a geometrical phase transition in non-equilibrium disordered systems [1]. As an occupation probability p is increased beyond a transition point p c , a macroscopicscale giant cluster emerges across the system. Theory of percolation transition was well established by the Kasteleyn-Fortuin formula [2]. This percolation theory has been used for understanding percolation-related diverse phenomena such as conductor-insulator transitions [3], the resilience of systems [4][5][6], the formation of public opinion [7,8], and the spread of disease in a population [9,10]. The percolation transition is known to be one of the most robust continuous transitions [1,11].Recently, however, many abrupt percolation transitions have been observed in complex systems [12][13][14][15][16][17][18], for instance, large-scale blackouts in power grid systems [19] and pandemics [20], in which the order parameter increases abruptly at a transition point. Among those transitions, an HPT has attracted substantial attention. The transitions in k-core percolation [21][22][23][24][25] and in the cascading failure model on interdependent networks [19,[26][27][28][29] are prototypical instances of the HPT. For these cases, the HPT is driven by cascade failures over the entire system as links are removed. The cluster size distribution (CSD) does not obey a power law. Instead, the avalanche size distribution follows a power law and shows critical behavior [29]. Consequently, the conventional formalism of percolation transition based on the CSD cannot be extended in an appropriate way to the HPT.Here, we aim to develop a theoretical framework of the critical phenomena of the HPT. To achieve this goal, we use a modified version [30] of the so-called half-restricted percolation model [31] in two and infinite dimensions. This model has potential applications to the transport or communication systems with global control equipments [30]. This model exhibits a HPT induced by cluster merging dynamics as links are added. The order parameter remains zero up to a transition point, at which it increases abruptly to a finite value, leading to a firstorder transition. As the order parameter abruptly increases,...

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Recent advances and open challenges in percolation

Cited by 132 publications

References 190 publications

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth

Critical behavior of a two-step contagion model with multiple seeds

Critical phenomena of a hybrid phase transition in cluster merging dynamics

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