Explosive Percolation in Erdős-Rényi-Like Random Graph Processes

Παναγιώτου, Κωνσταντίνος; Spöhel, Reto; Steger, Angelika; Thomas, Henning

doi:10.1016/j.endm.2011.10.017

Cited by 40 publications

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“…We will show here, that for a specific case, namely a variant of the model introduced in Ref. [3], critical fluctuations and power-law distributions can prevail and for the first time identify a hybrid transition in explosive percolation.…”

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confidence: 99%

“…The model is depicted schematically in Fig.1. We remark that this r-ER model is a slightly modified version of the original model [3] in which the number of nodes in the set R is always gN , independent of time. Thus, when N k−1 (t) < gN < N k (t), some nodes in one of the largest clusters in R belong to R and the other nodes of the same cluster belong to R (c) .…”

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Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents

et al. 2016

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Consider growing a network, in which every new connection is made between two disconnected nodes. At least one node is chosen randomly from a subset consisting of g fraction of the entire population in the smallest clusters. Here we show that this simple strategy for improving connection exhibits a phase transition barely studied before, namely a hybrid percolation transition exhibiting the properties of both first-order and second-order phase transitions. The cluster size distribution of finite clusters at a transition point exhibits power-law behavior with a continuously varying exponent τ in the range 2 < τ (g) ≤ 2.5. This pattern reveals a necessary condition for a hybrid transition in cluster aggregation processes, which is comparable to the power-law behavior of the avalanche size distribution arising in models with link-deleting processes in interdependent networks.PACS numbers: 64.60. De,64.60.ah,89.75.Da Transport or communication systems grow by adding new connections. Often certain constraints are imposed by society and if these constraint involve global knowledge about connectivity, the transition to a percolating system can become first order, as happens for instance when suppressing the spanning cluster, when imposing a cluster size [1] or when favoring the most disconnected sites [2]. Typically this effect is accompanied by the loss of critical scaling making these abrupt transitions less predictable and thus more dangerous. We will show here, that for a specific case, namely a variant of the model introduced in Ref.[3], critical fluctuations and power-law distributions can prevail and for the first time identify a hybrid transition in explosive percolation.Hybrid phase transitions have been observed recently in many complex network systems [4,5]; in these transitions, the order parameter m(t) exhibits behaviors of both first-order and second-order transitions simultaneously aswhere m 0 and r are constants and β is the critical exponent of the order parameter, and t is a control parameter. Examples of such behavior include k-core percolation [6,7], the cascading failure model on interdependent complex networks [8,9], and the Kuramoto synchronization model with a correlation between the natural frequencies and degrees of each node on complex networks [10,11], etc. For the models in [6-9], a critical behavior appears as nodes or links are deleted from a percolating cluster above the percolation threshold until reaching a transition point t c . As t is decreased infinitesimally further as 1/N beyond t c in finite systems, the order parameter decreases suddenly to zero and a first-order phase transition occurs. Thus, a hybrid phase transition occurs at t = t c in the thermodynamic limit. Next we recall discontinuous percolation transitions occurring in generalized contagion models [12][13][14]. Recent studies [15] of a generalized epidemic model [13] revealed that the discontinuous percolation transition turns out to be a hybrid percolation transition (HPT) represented by (1). For this case, a HPT ...

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Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents

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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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“…We use periodic boundary conditions in the simulations. We call this model the restricted percolation model with reference to the original name, the half-restricted percolation model [31]. We remark that our dynamic rule is slightly different from the original one in that a cluster on the boundary between the two sets in the original model is regarded as an element of set R in our model.…”

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confidence: 99%

“…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].…”

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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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Explosive Percolation in Erdős-Rényi-Like Random Graph Processes

Cited by 40 publications

References 12 publications

Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents

Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents

Deriving an underlying mechanism for discontinuous percolation

Critical phenomena of a hybrid phase transition in cluster merging dynamics

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