Optimal transport exponent in spatially embedded networks

Li, G.; Reis, Saulo D. S.; Moreira, André A.; Havlin, Shlomo; Stanley, H. Eugene; Andrade, José S.

doi:10.1103/physreve.87.042810

Cited by 49 publications

(53 citation statements)

References 36 publications

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“…(1) remains an open question. Moreover the PLN is a rich structure that opens many questions in planar networks, we intend to explore in a future work the optimal ρ that maximize transport properties [24][25][26][27] in the PLN. The relation (1) was introduced to create models that express a power law distribution for the probability of finding connected node according to the distance.…”

Section: Final Remarksmentioning

confidence: 99%

A planar network between short and long range behaviour

Moreira

Lucena

Corso

2015

Physica A: Statistical Mechanics and its Applications

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Section: Final Remarksmentioning

confidence: 99%

A planar network between short and long range behaviour

Moreira

Lucena

Corso

2015

Physica A: Statistical Mechanics and its Applications

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“…It is then claimed that this condition is optimal due to the presence of strong correlations between the structure of the long-range connections and the underlying lattice, leading to the formation of "information gradients" that allow the traveler to find the target. Later, it was shown that, by imposing a cost constraint to the long-range connections, results in α opt = d+ 1, for both local and global knowledge conditions [26].A question that naturally arises from these navigation studies is how efficient small-world networks are for transport phenomena that typically obey local conservation laws. Here we show that enhanced Laplacian flow properties can also be observed for networks built by adding long-range connections to an underlying regular lattice, in the same fashion as previously proposed for navigation through small-world geometries [24][25][26][27][28].…”

mentioning

confidence: 99%

“…Later, it was shown that, by imposing a cost constraint to the long-range connections, results in α opt = d+ 1, for both local and global knowledge conditions [26].…”

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confidence: 99%

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confidence: 99%

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Enhanced Flow in Small-World Networks

et al. 2014

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The small-world property is known to have a profound effect on the navigation efficiency of complex networks [J. M. Kleinberg, Nature 406, 845 (2000)]. Accordingly, the proper addition of shortcuts to a regular substrate can lead to the formation of a highly efficient structure for information propagation. Here we show that enhanced flow properties can also be observed in these complex topologies. Precisely, our model is a network built from an underlying regular lattice over which long-range connections are randomly added according to the probability, Pij ∼ r −α ij , where rij is the Manhattan distance between nodes i and j, and the exponent α is a controlling parameter. The mean two-point global conductance of the system is computed by considering that each link has a local conductance given by gij ∝ r −δ ij , where δ determines the extent of the geographical limitations (costs) on the long-range connections. Our results show that the best flow conditions are obtained for δ = 0 with α = 0, while for δ ≫ 1 the overall conductance always increases with α. For δ ≈ 1, α = d becomes the optimal exponent, where d is the topological dimension of the substrate. Interestingly, this exponent is identical to the one obtained for optimal navigation in small-world networks using decentralized algorithms.The Laplacian matrix operator is a general description for systems presenting two essential properties: (i) they obey local conservation of some load, and (ii) [10,11], among others. In these problems, one is frequently interested in the stationary state, where currents in each edge of a given network can be determined using local conservation laws. In the field of complex networks, in particular, the Laplacian operator has been employed as a conceptual approach for determining the nature of the community structure in the networks [12], in the context of network synchronization [13], as well as to study network flow [14][15][16][17][18][19].Given a regular network as an underlying substrate, it has been shown that the addition of a small set of random long-range links can greatly reduce the shortest paths among its sites. In particular, if the average shortest path ℓ s grows slowly with the network size N , typically when ℓ s ∼ log(N ), the network is called a small world [20][21][22]. If one considers the effect of constraining the allocation of the long-range connections with a probability decaying with the distance, P ij ∼ r −α ij , results in an effective dimensionality for the chemical distances that depends on the value of α [23]. For the case in which the regular underlying lattice is one-dimensional, the small-world behavior has only been detected for α < 2, with ℓ s reaching a minimum at α opt = 0 [23]. The two-dimensional case was also investigated [24], yielding similar results.The situation is much more complex if one does not have the global information of all the short-cuts present in the network. As a consequence, the traveler does not have a priori knowledge of the shortest path. Optimal navigation with loca...

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“…More recently, with development of the theory of complex networks, significant advances have been made in understanding these processes. Much of the studies were concerned with the dependence of navigation on the topological characteristics of complex networks [3][4][5][6][7], ranking of nodes in terms of network navigation [8,9] and the development of random search strategies [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Greedy reduction of navigation time in random search processes

Trpevski

Kocarev

2016

2016 24th Telecommunications Forum (TELFOR)

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Random search processes are instrumental in studying and understanding navigation properties of complex networks, food search strategies of animals, diffusion control of molecular processes in biological cells, and improving web search engines. An essential part of random search processes and their applications are various forms of (continuous or discrete time) random walk models. The efficiency of a random search strategy in complex networks is measured with the mean first passage time between two nodes or, more generally, with the mean first passage time between two subsets of the vertex set. In this paper we formulate a problem of adding a set of k links between the two subsets of the vertex set that optimally reduce the mean first passage time between the sets. We demonstrate that the mean first passage time between two sets is non-increasing and supermodular set function defined over the set of links between the two sets. This allows us to use two greedy algorithms that approximately solve the problem and we compare their performance against several standard link prediction algorithms. We find that the proposed greedy algorithms are better at choosing the links that reduce the navigation time between the two sets.

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Optimal transport exponent in spatially embedded networks

Cited by 49 publications

References 36 publications

A planar network between short and long range behaviour

A planar network between short and long range behaviour

Enhanced Flow in Small-World Networks

Greedy reduction of navigation time in random search processes

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