Universal Behavior of Optimal Paths in Weighted Networks with General Disorder

Havlin, Shlomo; Stanley, H. Eugene

doi:10.1103/physrevlett.96.068702

Cited by 42 publications

(40 citation statements)

References 18 publications

(42 reference statements)

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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…”

mentioning

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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mentioning

confidence: 82%

Stability of directed Min-Max optimal paths

Perlsman¹,

Havlin²

2007

Europhys. Lett.

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“…Determining ξ for a given distribution is addressed in [14], and we return to this in the discussion of results.…”

Section: Model and Methods -mentioning

confidence: 99%

“…In order to understand the previous results, we consider the current knowledge on the problem of optimal paths, which has received considerable attention in the context of surface growth and domain walls [14,15,21] in the physics literature. For the purpose of clarity we briefly review these results here, starting with lattices and extending the discussion to networks.…”

Section: Model and Methods -mentioning

confidence: 99%

“…Generally, we would expect that disorder induced LPP changes as a function of the form of P a (w). However, recent work [14] shows that large classes of disorder distributions are essentially equivalent in the optimal path problem, provided a certain characteristic length scale ξ(a) associated with P a (w) is conserved (see below). In practical terms, this means that disorder distributions of different functional forms but with equal ξ(a) lead to optimal path distributions that scale in the same way [14].…”

Section: Model and Methods -mentioning

confidence: 99%

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Disorder-induced Limited Path Percolation

Braunstein¹

2012

EPL

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Abstract. -We introduce a model of percolation induced by disorder, where an initially homogeneous network with links of equal weight is disordered by the introduction of heterogeneous weights for the links. We consider a pair of nodes i and j to be mutually reachable when the ratio αij of length of the optimal path between them before and after the introduction of disorder does not increase beyond a tolerance ratio τ . These conditions reflect practical limitations of reachability better than the usual percolation model, which entirely disregards path length when defining connectivity and, therefore, communication. We find that this model leads to a first order phase transition in both 2-dimensional lattices and in Erdős-Rényi networks, and in the case of the latter, the size of the discontinuity implies that the transition is effectively catastrophic, with almost all system pairs undergoing the change from reachable to unreachable. Using the theory of optimal path lengths under disorder, we are able to predict the percolation threshold. For real networks subject to changes while in operation, this model should perform better in predicting functional limits than current percolation models.Percolation theory is the most well-established approaches to study system connectivity, and how this connectivity becomes compromised with local system failure [1]. Systems usually refer to connected structures such as lattices or random networks [2], and failures to the removal of nodes or links. Percolation predicts failure thresholds and size of the connected parts of the network after those failures. Practical domains of applicationes include epidemiology [3][4][5], communication networks like the Internet [6], and propagation of information in social networks [7].

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“…Depending on the level of randomness on the weights of the connections, the system can fall into a strong-or a weak-disorder regime. The difference between these two regimes is that in the strong-disorder case the fluctuations on the weight are so large that the accumulated weight along each paths is controlled by the edge with the highest weight, while in weak-disorder the "responsibility" for the path's overall weight is distributed among all the links along the path [36,37]. In the two regimes, the scaling of b max , and of the average weight of the paths w path , with N changes with respect to unweighted graphs [25].…”

Section: Weighted Networkmentioning

confidence: 99%

Optimization of transport protocols with path-length constraints in complex networks

2010

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We propose a protocol optimization technique that is applicable to both weighted or unweighted graphs. Our aim is to explore by how much a small variation around the Shortest Path or Optimal Path protocols can enhance protocol performance. Such an optimization strategy can be necessary because even though some protocols can achieve very high traffic tolerance levels, this is commonly done by enlarging the path-lengths, which may jeopardize scalability. We use ideas borrowed from Extremal Optimization to guide our algorithm, which proves to be an effective technique. Our method exploits the degeneracy of the paths or their close-weight alternatives, which significantly improves the scalability of the protocols in comparison to Shortest Paths or Optimal Paths protocols, keeping at the same time almost intact the length or weight of the paths. This characteristic ensures that the optimized routing protocols are composed of paths that are quick to traverse, avoiding negative effects in data communication due to path-length increases that can become specially relevant when information losses are present.

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Universal Behavior of Optimal Paths in Weighted Networks with General Disorder

Cited by 42 publications

References 18 publications

Stability of directed Min-Max optimal paths

Stability of directed Min-Max optimal paths

Disorder-induced Limited Path Percolation

Optimization of transport protocols with path-length constraints in complex networks

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