Generating-function approach for bond percolation in hierarchical networks

Hasegawa, Takehisa; Satô, Masataka; Nemoto, Koji

doi:10.1103/physreve.82.046101

Cited by 27 publications

(74 citation statements)

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“…The phenomenology of percolation on hierarchical networks is quite distinct from that of lattices [11][12][13][14] . Specifically, on a lattice both, an end-to-end path and an extensive cluster, arise with certainty above the same critical bond density p c .…”

Section: Resultsmentioning

confidence: 99%

“…In particular, networks possessing hierarchical features 4,[6][7][8][9][10] relate to actual transport systems such as for air travel, routers and social interactions. Certain hierarchical networks, with a self-similar structure, have been shown to exhibit novel features 8,[11][12][13][14][15] . To these, we add here an unprecedented discontinuous transition in the formation of an extensive cluster for ordinary, random-bond addition.…”

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confidence: 99%

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Ordinary percolation with discontinuous transitions

2012

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Percolation on a one-dimensional lattice and fractals, such as the sierpinski gasket, is typically considered to be trivial, because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a non-trivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here we provide a simple example in the form of a small-world network consisting of a one-dimensional lattice which, when combined with a hierarchy of long-range bonds, reveals many features of this transition in a mathematically rigorous manner.

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Section: Resultsmentioning

confidence: 99%

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confidence: 99%

Ordinary percolation with discontinuous transitions

2012

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“…Here s max (N ; p) is the mean size of the largest cluster in a graph of size N at a given value of p. As known, p c1 = p c2 on Euclidean lattices, whereas, p c1 < p c2 on transitive nonamenable graphs [21,22]. Also, in complex networks, some growing network models [25][26][27][28][29] and hierarchical network models [23,[30][31][32][33] yield 0 = p c1 < p c2 for bond percolation, whereas, p c1 = p c2 for some static network models, such as uncorrelated networks…”

Section: Introductionmentioning

confidence: 99%

Hierarchical scale-free network is fragile against random failure

Hasegawa

Nemoto

2013

Phys. Rev. E

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We investigate site percolation in a hierarchical scale-free network known as the Dorogovtsev-Goltsev-Mendes network. We use the generating function method to show that the percolation threshold is 1, i.e., the system is not in the percolating phase when the occupation probability is less than 1. The present result is contrasted to bond percolation in the same network of which the percolation threshold is zero. We also show that the percolation threshold of intentional attacks is 1. Our results suggest that this hierarchical scale-free network is very fragile against both random failure and intentional attacks. Such a structural defect is common in many hierarchical network models.

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“…A critical phase, if it exists, lies between a disordered phase withξ < 1/y d and an ordered phase withξ = ∞. Such a phase with a fractal exponent 0 < ψ < 1 is actually observed in the percolation transitions in enhanced trees [5], hyperbolic lattices [10], hierarchical graphs [11,12], and growing random networks [13].…”

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Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

2012

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We study the Ising model in a hierarchical small-world network by renormalization group analysis and find a phase transition between an ordered phase and a critical phase, which is driven by the coupling strength of the shortcut edges. Unlike ordinary phase transitions, which are related to unstable renormalization fixed points (FPs), the singularity in the ordered phase of the present model is governed by the FP that coincides with the stable FP of the ordered phase. The weak stability of the FP yields peculiar criticalities, including logarithmic behavior. On the other hand, the critical phase is related to a nontrivial FP, which depends on the coupling strength and is continuously connected to the ordered FP at the transition point. We show that this continuity indicates the existence of a finite correlation-length-like quantity inside the critical phase, which diverges upon approaching the transition point. Recently, various physical phenomena in non-Euclidean graphs have been studied, especially in the context of complex networks, and their properties have been found to be beyond the scope of the conventional theory for Euclidean graphs [1]. Of particular interest are systems regarded as infinitedimensional in the sense that the equidistant surface S r of radius r grows exponentially as S r ∝ e y d r with a positive constant y d , which is faster than any power function r d−1 as in d-dimensional Euclidean graphs. Typical examples are trees and hyperbolic lattices [2]. Remarkably, such infinitedimensional systems often exhibit the critical phases in which the (nonlinear) susceptibility diverges [3][4][5]. Although the critical phase is also observed in some Euclidean systems, e.g., the quasi-long-range ordered phase in the two-dimensional XY model [6,7], the critical phases in infinite-dimensional systems are considered to be due to rather geometrical effects. Indeed, the exponential growth of a graph admits the divergence of the susceptibility χ , which is calculated by the integral of the two-point correlation function [8], even with finite correlation lengthξ [9] The property of the phase transitions between a critical phase and an ordered phase is an interesting issue. Quite recently, some models in the simple hierarchical network shown in Fig. 1(a) were investigated to examine these transitions. This network is very useful because rigorous real-space renormalization is possible for various models in the simplest way. Furthermore various types of phase transition are observed depending on the model used, e.g., a discontinuous transition of the bond percolation model [12], equivalent to the one-state Potts model [14,15], and continuous transitions with a power-law singularity (PLS) or an essential singularity (ES) for the q-state Potts model with q 3 [16]. These are observed in other graphs [3][4][5]17,18]. Thus this hierarchical network is a good stage to investigate what determines the type of phase transitions in a systematic way. In particular the two-state Potts model, equivalent to the Ising model, ...

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Generating-function approach for bond percolation in hierarchical networks

Cited by 27 publications

References 27 publications

Ordinary percolation with discontinuous transitions

Ordinary percolation with discontinuous transitions

Hierarchical scale-free network is fragile against random failure

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

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