Optimal paths in disordered media: Scaling of the crossover from self-similar to self-affine behavior

Porto, Markus; Schwartz, Nehemia; Havlin, Shlomo; Bunde, Armin

doi:10.1103/physreve.60.r2448

Cited by 48 publications

(62 citation statements)

References 20 publications

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“…Here we derive analytically such a criterion, and support our derivation by extensive simulations. Using this criterion we show that certain power law distributions and lognormal distributions P (w) can lead to strong disorder and to a weak-strong disorder crossover [12]. We also show that for P (w) uniform, Poisson or Gaussian, only weak disorder occurs regardless of the broadness of P (w).…”

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confidence: 95%

“…with d opt = 1.22 (d = 2) and d opt = 1.42 (d = 3) for lattices [2,12], and ℓ ∼ N 1/3 for random networks [7]. It is commonly agreed that strong disorder arises only when P (w) is broad enough.…”

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confidence: 99%

“…To identify x 1 and x 2 in (5), we show that the strong disorder case is related to percolation [12]. The percolation problem corresponds to the construction of random networks by the removal of links with probability 1−p.…”

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confidence: 99%

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Universal Behavior of Optimal Paths in Weighted Networks with General Disorder

Havlin

Stanley

2006

Phys. Rev. Lett.

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We study the statistics of the optimal path in both random and scale free networks, where weights w are taken from a general distribution P (w). We find that different types of disorder lead to the same universal behavior. Specifically, we find that a single parameter (S ≡ AL −1/ν for d-dimensional lattices, and S ≡ AN −1/3 for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here ν is the percolation connectivity exponent, and A depends on the percolation threshold and P (w). For P (w) uniform, Poisson or Gaussian the crossover from weak to strong does not occur, and only weak disorder exists.

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Universal Behavior of Optimal Paths in Weighted Networks with General Disorder

Havlin

Stanley

2006

Phys. Rev. Lett.

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“…00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 11111111111111111111111111 by N νopt , where ν opt = 1/3 for ER and for SF with λ > 4, and ν opt = (λ − 3)/(λ − 1) for SF networks with 3 < λ < 4 [9]. For the L × L square lattice, ℓ MST ∼ L dopt and since L 2 = N , ν opt = d opt /2 ≈ 0.61 [7,8]. …”

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confidence: 99%

Transport in Weighted Networks: Partition into Superhighways and Roads

Wu¹,

Braunstein²,

Havlin³

et al. 2006

Phys. Rev. Lett.

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Transport in weighted networks is dominated by the minimum spanning tree (MST), the tree connecting all nodes with the minimum total weight. We find that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality -number of times a node (or link) is used by transport paths. One component, the superhighways, is the infinite incipient percolation cluster; for which we find that nodes (or links) with high centrality dominate. For the other component, roads, which includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. The significance of this finding by showing that one can improve significantly the global transport by improving a very small fraction of the network, the superhighways.

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“…The second case is the one of strong disorder: While in the strong disorder limit the regular optimal path is identical to the Min-Max optimal path, as the strength of the disorder decreases, a strong disorder -weak disorder transition occurs. This transition was studied for the ordinary (non-directed) lattice in [14,15], and for the directed case in [16]. The present study shows that in the case of small perturbations, the probability to depart from the original Min-Max optimal path is…”

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confidence: 82%

Stability of directed Min-Max optimal paths

Perlsman¹,

Havlin²

2007

Europhys. Lett.

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-The stability of directed Min-Max optimal paths in cases of change in the random media is studied. Using analytical arguments it is shown that when small perturbations are applied to the weights of the bonds of the lattice, the probability that the new Min-Max optimal path is different from the original Min-Max optimal path is proportional to t 1/ν , where t is the size of the lattice, and ν is the longitudinal correlation exponent of the directed percolation model. It is also shown that in a lattice whose bonds are assigned with weights which are near the strong disorder limit, the probability that the directed polymer optimal path is different from the optimal Min-Max path is proportional to t 2/ν /k 2 , where k is the strength of the disorder. These results are supported by numerical simulations. Copyright c EPLA, 2007The directed polymer model [1] is a well-studied [2] model in the field of disordered systems. The model is concerned with directed optimal paths in random media, which are characterized by two growth rate exponents, ω and ν. The exponent ω determines the energy variability of a path of length t by the relation ∆E ∼ t ω , and the exponent ν determines the mean transversal distance of the optimal paths from the origin, through the relation D ∼ t ν . These two growth rate exponents are connected by the Huse-Henley scaling relation: ω = 2ν − 1 [3]. There are two cases in which the space exponent ν has the value of the directed percolation model [4], rather than its value in the regular case. In the first case [5,6], the bonds of the lattice are assigned with values taken from a bimodal (0,1) distribution, and the probability to have a zero-valued bond is p c , the critical probability of directed percolation. In the second case, the bonds are assigned with values taken from a strong disordered distribution, and thus the energy of each path, which is defined as the sum of its bonds' values, is mainly determined by the value of the maximal bond along that path. In the strong disorder limit, the optimal paths are identical to the optimal Min-Max paths, which are characterized by the directed percolation exponent ν [7,8].Directed Min-Max optimal paths are the subject of the present article, which studies two cases in which there is a small probability for a change in the position of the optimal path. In the first case, small perturbations are applied to the weights of the bonds of the lattice, and the new Min-Max optimal path might be different from the original one. This case was numerically studied in [9] for directed polymer (regular) optimal paths (which are determined by the minimal sum of bond values), and a general theoretical discussion was presented in [10]. Explanations to the numerical results presented in [9] were given in [10][11][12][13]. The second case is the one of strong disorder: While in the strong disorder limit the regular optimal path is identical to the Min-Max optimal path, as the strength of the disorder decreases, a strong disorder -weak disorder transition occurs. ...

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Optimal paths in disordered media: Scaling of the crossover from self-similar to self-affine behavior

Cited by 48 publications

References 20 publications

Universal Behavior of Optimal Paths in Weighted Networks with General Disorder

Universal Behavior of Optimal Paths in Weighted Networks with General Disorder

Transport in Weighted Networks: Partition into Superhighways and Roads

Stability of directed Min-Max optimal paths

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