Scale-Free Networks on Lattices

Alejandro F. Rozenfeld, Reuven Cohen, Daniel ben-Avraham, and Shlomo Havlin

Phys. Rev. Lett. 89, 218701 – Published 1 November 2002

Abstract

We suggest a method for embedding scale-free networks, with degree distribution $P (k) \sim k^{- λ}$ , in regular Euclidean lattices accounting for geographical properties. The embedding is driven by a natural constraint of minimization of the total length of the links in the system. We find that all networks with $λ > 2$ can be successfully embedded up to a (Euclidean) distance $ξ$ which can be made as large as desired upon the changing of an external parameter. Clusters of successive chemical shells are found to be compact (the fractal dimension is $d_{f} = d$ ), while the dimension of the shortest path between any two sites is smaller than 1: $d_{m i n} = (λ - 2) / (λ - 1 - 1 / d)$ , contrary to all other known examples of fractals and disordered lattices.

Received 31 May 2002

DOI:https://doi.org/10.1103/PhysRevLett.89.218701

Authors & Affiliations

Alejandro F. Rozenfeld¹, Reuven Cohen¹, Daniel ben-Avraham², and Shlomo Havlin¹

¹Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
²Department of Physics, Clarkson University, Potsdam, New York 13699-5820

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Issue

Vol. 89, Iss. 21 — 18 November 2002

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Images

Figure 1
Spatial structure of connectivity network. (a) Shown is the typical map of links for a system of $50 \times 50$ sites generated from connectivity distributions with $λ = 2.5$ and $λ = 5$ . (b) Shown are shells of equidistant sites to the central one in a lattice of $300 \times 300$ sites. Note that, for $λ = 5$ , shells are concentric and continuous fields; but for $λ = 2.5$ shells are broken.Reuse & Permissions
Figure 2
(a) The resulting connectivity distribution obtained from simulations performed on two dimensional systems of size $R = 400$ , $λ = 2.5$ , and for several values of $A$ : (circles) $A = 2$ , (squares) $A = 3$ , and (diamonds) $A = 4$ ; they all end at a cutoff $k_{c} (A)$ . For this case $r_{m a x} > ξ$ . In the inset, we show scaling collapse using the same data. The threshold takes place at $k_{c} \sim \frac{1}{λ - 2} A^{d}$ and confirms the validity of our theoretical estimations. (b) Power-law distribution of site connectivity in the network is shown for $R = 100$ , $A = 10$ , and for different values of $λ$ : $λ = 2.5$ (circles), 3.0 (squares), and 5.0 (diamonds). Note that in all cases the distribution achieves its (natural) cutoff $K$ . In the inset, we show the corresponding collapse supporting $K \sim R^{d / λ - 1}$ . For this case, $r_{m a x} < ξ$ .Reuse & Permissions
Figure 3
This diagram shows the six regions where different behavior of the network is found: For region A: $r_{m a x} < R < ξ$ ; B: $r_{m a x} < ξ < R$ ; C: $ξ < r_{m a x} < R$ ; D: $ξ < R < r_{m a x}$ ; E: $R < ξ < r_{m a x}$ ; F: $R < r_{m a x} < ξ$ . The diagram can be mapped into only four regions where the cutoff $k_{c}$ and where size effect $K$ are expected. A and B: no cutoff and no size effect; C and D: cutoff and no size effect; E: cutoff and size effect; F: no cutoff but size effect. The two symbols indicate the parameters corresponding to Fig. 1b, ( full diamond) $λ = 2.5$ and (full circle) $λ = 5$ .Reuse & Permissions
Figure 4
Plot of scaled perimeter as a function of the Euclidean distance from the central site, for several values of $λ$ : $λ = 3.0$ (top), 3.25, 3.5, 3.75, and 4.0 (bottom). The simulations where performed with $A = 7$ . Note that the position where the curves split, $r ≃ A$ , is consistent with our analytical results [Eq. (7)]. Also, the asymptotic values shown for large $r$ are consistent with $c (λ) A$ .Reuse & Permissions
Figure 5
(a) The minimal length exponent $d_{m i n}$ as a function of $λ$ . Note the good agreement between theoretical estimations (continuous line) and simulations results (full squares). (b) The shape of the $Φ (l^{d_{l}} / R^{d})$ scaling function is shown for $λ = 4$ and several lattice sizes: $R = 1000$ (circle), 2000 (square), 2500 (diamond), and 3000 (triangle).Reuse & Permissions

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