Tricritical Point in Heterogeneous<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>k</mml:mi></mml:math>-Core Percolation

Cellai, Davide; Lawlor, Aonghus; Dawson, Kenneth A.; Gleeson, James P.

doi:10.1103/physrevlett.107.175703

Cited by 65 publications

(77 citation statements)

References 32 publications

(49 reference statements)

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“…There are no tricritical points in these systems, but it was shown in [29] that a tricritical point does appear in the (2,3) heterogeneous k core. Noting that a similar tricritical point appears in partial interdependence models [18,19], it seems natural to consider WPP and the partial interdependence model as specific cases of a broader class of mixed-rule multiplex percolation models.…”

Section: Discussionmentioning

confidence: 99%

Weak percolation on multiplex networks

et al. 2014

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Bootstrap percolation is a simple but nontrivial model. It has applications in many areas of science and has been explored on random networks for several decades. In single-layer (simplex) networks, it has been recently observed that bootstrap percolation, which is defined as an incremental process, can be seen as the opposite of pruning percolation, where nodes are removed according to a connectivity rule. Here we propose models of both bootstrap and pruning percolation for multiplex networks. We collectively refer to these two models with the concept of "weak" percolation, to distinguish them from the somewhat classical concept of ordinary ("strong") percolation. While the two models coincide in simplex networks, we show that they decouple when considering multiplexes, giving rise to a wealth of critical phenomena. Our bootstrap model constitutes the simplest example of a contagion process on a multiplex network and has potential applications in critical infrastructure recovery and information security. Moreover, we show that our pruning percolation model may provide a way to diagnose missing layers in a multiplex network. Finally, our analytical approach allows us to calculate critical behavior and characterize critical clusters.

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Section: Discussionmentioning

confidence: 99%

Weak percolation on multiplex networks

et al. 2014

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“…Recently, k-core (KC) percolation [5][6][7][8] and heterogeneous k-core (HKC) percolation [9][10][11][12][13] have been intensively studied. As the generalized concept of the giant component, KC gives a deeper insight into the structure and organization of complex networks [10,11]. KC percolation is relevant to understanding the resilience of a network under random damage [12].…”

Section: Introductionmentioning

confidence: 99%

“…HKC on a network is also defined as the maximal cluster in which each node i has its own threshold k i and has at least k i directly linked neighbors within the cluster itself [10]. Moreover, HKC percolation with generalized thresholds was also found to have more wide applications to various fields, such as * syook@khu.ac.kr † Corresponding author: ykim@khu.ac.kr opinion formations [18], epidemic spreading [19], robustness of network [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Complete set of types of phase transition in generalized heterogeneousk-core percolation

2014

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We study heterogeneous k-core (HKC) percolation with a general mixture of the threshold k, with k(min) = 2 on random networks. Based on the local tree approximation, the scaling behaviors of the percolation order parameter P(∞)(p) are analytically obtained for general distributions of the threshold k. The analytic calculations predict that the generalized HKC percolation is completely described by the series of continuous transitions with order parameter exponents β(n) = 2/n, discontinuous hybrid transitions with β(H) = 1/2 or β(A)(4)) = 1/4, and three kinds of multiple transitions. Simulations of the generalized HKC percolations are carried out to confirm analytically predicted transition natures. Specifically, the exponents of the series of continuous transitions are shown to satisfy the hyperscaling relation 2β(n) + γ(n) = ν(n).

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“…Several models that lead to truly discontinuous percolation transitions are now known, e.g. [14][15][16][17][18][19][20][21], yet the underlying mechanisms are not fully understood. There are many investigations underway to isolate essential ingredients that lead to a discontinuous transition such as cooperative phenomena [22], hierarchical structures [21], correlated percolation [23], and algorithms that explicitly suppress types of growth [24].…”

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Deriving an underlying mechanism for discontinuous percolation

Chen¹,

Zheng²,

D’Souza³

2012

EPL

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-Understanding what types of phenomena lead to discontinuous phase transitions in the connectivity of random networks is an outstanding challenge. Here we show that a simple stochastic model of graph evolution leads to a discontinuous percolation transition and we derive the underlying mechanism responsible: growth by overtaking. Starting from a collection of n isolated nodes, potential edges chosen uniformly at random from the complete graph are examined one at a time while a cap, k, on the maximum allowed component size is enforced. Edges whose addition would exceed k can be simply rejected provided the accepted fraction of edges never becomes smaller than a function which decreases with k as g(k) = 1/2 + (2k) −β . We show that if β < 1 it is always possible to reject a sampled edge and the growth in the largest component is dominated by an overtaking mechanism leading to a discontinuous transition. If β > 1, once k ≥ n 1/β , there are situations when a sampled edge must be accepted leading to direct growth dominated by stochastic fluctuations and a "weakly" discontinuous transition. We also show that the distribution of component sizes and the evolution of component sizes are distinct from those previously observed and show no finite size effects for the range of β studied.

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Tricritical Point in Heterogeneousk-Core Percolation

Cited by 65 publications

References 32 publications

Weak percolation on multiplex networks

Weak percolation on multiplex networks

Complete set of types of phase transition in generalized heterogeneousk-core percolation

Deriving an underlying mechanism for discontinuous percolation

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