Optimal transport exponent in spatially embedded networks

G. Li, S. D. S. Reis, A. A. Moreira, S. Havlin, H. E. Stanley, and J. S. Andrade, Jr.

Phys. Rev. E 87, 042810 – Published 18 April 2013

Abstract

The imposition of a cost constraint for constructing the optimal navigation structure surely represents a crucial ingredient in the design and development of any realistic navigation network. Previous works have focused on optimal transport in small-world networks built from two-dimensional lattices by adding long-range connections with Manhattan length $r_{i j}$ taken from the distribution $P_{i j} \sim r_{i j}^{- α}$ , where $α$ is a variable exponent. It has been shown that, by introducing a cost constraint on the total length of the additional links, regardless of the strategy used by the traveler (independent of whether it is based on local or global knowledge of the network structure), the best transportation condition is obtained with an exponent $α = d + 1$ , where $d$ is the dimension of the underlying lattice. Here we present further support, through a high-performance real-time algorithm, on the validity of this conjecture in three-dimensional regular as well as in two-dimensional critical percolation clusters. Our results clearly indicate that cost constraint in the navigation problem provides a proper theoretical framework to justify the evolving topologies of real complex network structures, as recently demonstrated for the networks of the US airports and the human brain activity.

Received 30 August 2012

DOI:https://doi.org/10.1103/PhysRevE.87.042810

Authors & Affiliations

G. Li¹, S. D. S. Reis², A. A. Moreira², S. Havlin^1,3, H. E. Stanley¹, and J. S. Andrade, Jr.²

¹Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
²Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
³Department of Physics, Bar-Ilan University, 52900 Ramal-Gab, Israel

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Vol. 87, Iss. 4 — April 2013

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Images

Figure 1
Two-dimensional square lattice with long-range connections. Each node has four short-range connections to its nearest neighbors. A long-range connection is added to a randomly chosen node $i$ with probability proportional to $r^{- α}$ . Here $r = 2$ , there are eight nodes (on the dashed square box) with the same lattice distance $r$ to node $i$ , and we randomly choose the node $j$ from these eight nodes to be connected to node $i$ .Reuse & Permissions
Figure 2
A fraction of 10 $%$ of nodes in a regular $d$ -dimensional ( $d = 1$ , 2, and 3) lattice with linear size $L = 1000$ are randomly selected to receive long-range connections with different $α$ . As seen, the optimal $〈 ℓ 〉$ is achieved for $α = 0$ . The results are averaged over 4000 realizations for each network. Note that when $α$ increases above the value of $d$ , $〈 ℓ 〉$ increases dramatically. For the dependence of $〈 ℓ 〉$ on $L$ for different $α$ , see Refs. [28, 32, 34].Reuse & Permissions
Figure 3
(a) Distribution and (b) normalized distribution of the shortest path length $ℓ$ for a two-dimensional lattice ( $L = 1000$ ) with additional long-range connections where the total length $Λ$ of the added long-range connections is limited to $N = L^{2}$ . Note the nonmonotonic behavior with respect to $α$ . For $α = d + 1 = 3$ the location of the peak of the distribution is the smallest. In (b), $〈 ℓ 〉$ is the mean $ℓ$ and $σ$ is the standard deviation for each curve. We sampled 100 000 network realizations for each $α$ .Reuse & Permissions
Figure 4
Average shortest path length $〈 ℓ 〉$ as a function of $α$ for (a) one-, (b) two-, and (c) three-dimensional lattices and (d) a fractal ( $d = d_{f} ≅ 1.9$ ) with additional long-range connections taken from the power-law distribution (1) as a function of $α$ . The total length $Λ$ of the added long-range connections is limited to $10 N$ for the one-dimensional lattice, $N$ for the two- and three-dimensional lattices, and $N_{f}$ for the fractal. The plots suggest that the optimal shortest path length is achieved at $α = d + 1$ for regular lattices and $α = d_{f} + 1$ for the fractal. Note that (b) is similar to Fig. 3 in Ref. [27] but with larger system sizes. The results are averaged over 4000 realizations for the three smaller $L$ and 400 realizations for the largest $L$ .Reuse & Permissions
Figure 5
Average shortest path length $〈 ℓ 〉$ as a function of system linear size $L$ with different $α$ for (a) one-, (b) two-, (c) and three-dimensional lattices and (d) a fractal ( $d_{f} ≅ 1.9$ ) with additional long-range connections taken from the power-law distribution (1). The total length $Λ$ of the added long-range connections is limited to $10 N$ for the one-dimensional lattice, $N$ for the two- and three-dimensional lattices, and $N_{f}$ for the fractal. The plots suggest that the optimal shortest path length is achieved at $α = d + 1$ for regular lattices and $α = d_{f} + 1$ for the fractal. For $d = 1$ , the slope of the fitting line $δ_{s} ≅ 0.54$ for $α = d + 1 = 2$ , $δ_{s} \approx 0.84$ for $α = 1.5$ and $2.5$ , and $δ_{s} \approx 1$ for $α = 0$ , 1, and 3. For $d = 2$ , $δ_{s} \approx 0.60$ for $α = 0$ and 1 and $δ ≅ 0.71$ for $α = 4$ ; however, for $α = d + 1 = 3$ , $〈 ℓ 〉$ seems to follow a weaker dependence from a power law, more likely a logarithmic law [see Fig. 6 (b)]. For $d = 3$ , $δ_{s} \approx 0.46$ for $α = 0, 1$ , and 2, $δ_{s} ≅ 0.40$ for $α = 3$ , and $δ_{s} ≅ 0.45$ for $α = 5$ ; however, for $α = d + 1 = 4$ , $〈 ℓ 〉$ seems to follow a logarithmic law. For $d = 1.9$ , $δ_{s} \approx 0.75$ for $α = 0$ and 1 and $δ ≅ 0.89$ for $α = 4$ ; however, for $α = d_{f} + 1 = 2.9$ , $〈 ℓ 〉$ seems to follow a logarithmic law. The results are averaged over 4000 realizations for each $α$ for $d = 1, 2$ , and 3 and 1000 realizations for $d = 1.9$ .Reuse & Permissions
Figure 6
Successive slopes for $d = 2$ of (a) $δ_{s}$ obtained from $\ln 〈 ℓ 〉$ vs $\ln L$ [of Fig. 5b], (b) $〈 ℓ 〉$ as a function of $\ln L$ in a double logarithmic plot, and (c) $γ_{s}$ obtained from $\ln 〈 ℓ 〉$ vs $\ln \ln L$ taken from (b). The total length $Λ$ of the added long-range connections is limited to $N = L^{2}$ . Note that in (a) for $α = d + 1 = 3$ , $δ_{s}$ decreases with $L$ , while for other values of $α$ , $δ_{s}$ is roughly constant. In (c) for $α = d + 1 = 3$ , $γ_{s}$ keeps roughly constant and for other values of $α$ , $γ_{s}$ increases with $L$ . This suggests that for $α = 3$ the relation between $〈 ℓ 〉$ and $L$ is a function that increases less than a power law and more likely that $〈 ℓ 〉$ increases logarithmically with $L$ .Reuse & Permissions

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