Asymmetric evolving random networks

Coulomb, Stephane; Bauer, Michel

doi:10.1140/epjb/e2003-00290-4

Cited by 28 publications

(37 citation statements)

References 9 publications

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“…It is only atp =p c that the percolation on the decorated (2,2)-flower shows an infinite order transition with the BKT-like singularity as percolations on growing networks do [7][8][9][10]. The finiteness of β forp <p c suggests that the existence of some critical phase adjacent to the normal ordered phase is not enough for the network to have such an essential singularity in the order parameter and thus an infinite order phase transition.…”

Section: Discussionmentioning

confidence: 99%

“…Note that (3) is re-obtained by putting x = 1 to (9) and using (10). The mean number of the root cluster s 0 n (or the order parameter P (n) ∞ (p)) and the cluster size distribution n (n) s (p) onF n are given as…”

Section: Generating Functionsmentioning

confidence: 99%

“…in a finite region below the transition point where no giant component exists [7][8][9][10][11]. A similar non-ordered phase with some power-law behavior, a critical phase, has also been observed in bond percolations on the enhanced binary tree [17][18][19], which is one of nonamenable graphs * Electronic address: hasegawa@stat.t.u-tokyo.ac.jp † Electronic address: hijiri@statphys.sci.hokudai.ac.jp ‡ Electronic address: nemoto@statphys.sci.hokudai.ac.jp (NAGs) [20,21].…”

Section: Introductionmentioning

confidence: 99%

“…The scaling relation (2) indicates that ψ L plays a role of the natural cut-off exponent of n s as shown in the growing random tree in which p c1 = 0 and p c2 = 1 [11]. In general the growing random networks are considered to have p c1 = 0 with finite p c2 [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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Generating-function approach for bond percolation in hierarchical networks

2010

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We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcutsp and that of the ordinary bonds p. The system has a critical phase in which the percolating probability P takes an intermediate value 0 < P < 1. Using generating function approach, we calculate the fractal exponent ψ of the root clusters to show that ψ varies continuously withp in the critical phase. We confirm numerically that the distribution ns of cluster size s in the critical phase obeys a power law ns ∝ s −τ , where τ satisfies the scaling relation τ = 1 + ψ −1 . In addition the critical exponent β(p) of the order parameter varies asp, from β ≃ 0.164694 atp = 0 to infinity atp =pc = 5/32.

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Section: Discussionmentioning

confidence: 99%

Section: Generating Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generating-function approach for bond percolation in hierarchical networks

2010

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“…An infinite-order percolation transition has been seen in models of growing networks [31,32,33,34,35,36,37,38], with exponential scaling in the size of the giant component above the percolation threshold. A prior observation of a finite-temperature, inverted Berezinskii-KosterlitzThouless singularity similar to the one described above was made in Ising models on an inhomogeneous growing network [39] and on a one-dimensional inhomogeneous lattice [40,41,42,43].…”

Section: Introductionmentioning

confidence: 99%

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

2006

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We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks-a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability p of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly-clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution P (k) and clustering coefficient C for all p, as well as the average path length ℓ for p = 0 and 1. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to 562,500 renormalized probability bins to represent the distribution. For p < 0.494, we find powerlaw critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with p, and exponential decay of correlations away from Tc. For p ≥ 0.494, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with non-zero magnetization and finite correlation length and a hightemperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching Tc from below, the magnetization and the susceptibility respectively exhibit the singularities of exp(−C/ √ Tc − T ) and exp(D/ √ Tc − T ), with C and D positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all p, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

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Growth Models for Networks

Dorogovtsev

2009

Encyclopedia of Complexity and Systems Science

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Asymmetric evolving random networks

Cited by 28 publications

References 9 publications

Generating-function approach for bond percolation in hierarchical networks

Generating-function approach for bond percolation in hierarchical networks

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

Growth Models for Networks

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