Degree heterogeneity in spatial networks with total cost constraint

📝 Abstract

Recently, In [Phys. Rev. Lett. 104, 018701 (2010)] the authors studied a spatial network which is constructed from a regular lattice by adding long-range edges (shortcuts) with probability $P_{ij}\sim r_{ij}^{-\alpha} $, where $r_{ij}$ is the Manhattan length of the long-range edges. The total length of the additional edges is subject to a cost constraint ( $\sum r=C $). These networks have fixed optimal exponent $\alpha$ for transportation (measured by the average shortest-path length). However, we observe that the degree in such spatial networks is homogenously distributed, which is far different from real networks such as airline systems. In this paper, we propose a method to introduce degree heterogeneity in spatial networks with total cost constraint. Results show that with degree heterogeneity the optimal exponent shifts to a smaller value and the average shortest-path length can further decrease. Moreover, we consider the synchronization on the spatial networks and related results are discussed. Our new model may better reproduce the features of many real transportation systems.

💡 Analysis

Recently, In [Phys. Rev. Lett. 104, 018701 (2010)] the authors studied a spatial network which is constructed from a regular lattice by adding long-range edges (shortcuts) with probability $P_{ij}\sim r_{ij}^{-\alpha} $, where $r_{ij}$ is the Manhattan length of the long-range edges. The total length of the additional edges is subject to a cost constraint ( $\sum r=C $). These networks have fixed optimal exponent $\alpha$ for transportation (measured by the average shortest-path length). However, we observe that the degree in such spatial networks is homogenously distributed, which is far different from real networks such as airline systems. In this paper, we propose a method to introduce degree heterogeneity in spatial networks with total cost constraint. Results show that with degree heterogeneity the optimal exponent shifts to a smaller value and the average shortest-path length can further decrease. Moreover, we consider the synchronization on the spatial networks and related results are discussed. Our new model may better reproduce the features of many real transportation systems.

📄 Content

arXiv:1112.0241v1 [physics.soc-ph] 1 Dec 2011 Degree heterogeneity in spatial networks with total cost constraint Weiping Liu, An Zeng,∗and Yanbo Zhou Department of Physics, University of Fribourg, Chemin du Mus´ee 3, CH-1700 Fribourg, Switzerland (Dated: July 27, 2021) Recently, In [Phys. Rev. Lett. 104, 018701 (2010)] the authors studied a spatial network which is constructed from a regular lattice by adding long-range edges (shortcuts) with probability Pij ∼r−α ij , where rij is the Manhattan length of the long-range edges. The total length of the additional edges is subject to a cost constraint (P r = C). These networks have fixed optimal exponent α for transportation (measured by the average shortest-path length). However, we observe that the degree in such spatial networks is homogenously distributed, which is far different from real networks such as airline systems. In this paper, we propose a method to introduce degree heterogeneity in spatial networks with total cost constraint. Results show that with degree heterogeneity the optimal exponent shifts to a smaller value and the average shortest-path length can further decrease. Moreover, we consider the synchronization on the spatial networks and related results are discussed. Our new model may better reproduce the features of many real transportation systems. PACS numbers: 89.75.Hc, 02.50.-r, 05.40.Fb, 89.75.Fb Spatial features play a significant role in the trans- portation networks [1], Internet [2], mobile phone net- works [3], power grids [4], social networks [5] and neural networks [6]. In the past decade, many systems have been modeled by complex networks where nodes and links are embedded in space. In these models, nodes are located in the plane and the geometric distance between nodes is well defined. Then links are constructed according to rules based on spatial indices [7–12]. Growing spatial net- works were also studied [13, 14]. Generally, these model of spatial networks are able to reproduce some properties of the real systems such as community structure, scale- free connection length distribution and so on. For a detail review of the field, see [15]. The most important features of spatial network model is that there is a cost associated with the length of links, which has dramatic effects on the topology and function of these networks. A total cost constraint has been re- cently introduced to design the spatial networks [16–21]. The total cost C is defined as the total length of the links, C = P r where r is the length of links. In [16], pairs of sites ij in 2-dimensional lattices are randomly chosen to be linked with long-range connections with probability Pij ∼r−α ij , where rij is the Manhattan distance between sites i and j. New links are added until the total length of the links reach the total cost C. The exponent α con- trols the trade-offbetween the link length and link num- ber. A large value of α allows for the formation of many short-length links while the small α favors the creation of a few long-length links. The authors in [16] show that the optimal exponents for both the average shortest-path length and navigation steps are α = 3 in 2-dimension and α = 2 in 1-dimension (the 1-dimensional scenario has been strictly proved very recently in [19]). The au- thors claim that such model reveals the optimized aspect of airline networks under the conditions of geographical ∗an.zeng@unifr.ch availability (for customer satisfaction) and cost limita- tions (for airline company profit). Similar works have been carried out to study dynamics such as traffic con- gestion and synchronization on spatial networks under total cost constraint, and the optimal link length expo- nents are found [18, 20, 21]. However, we observe that the degree distribution in these spatial network models with total cost constraint is homogenous, which is far different from real cases, in- cluding airline systems. Many previous works have re- vealed that many geographic-based networks have het- erogenous degree distribution [1]. In this paper, we pro- pose a method to introduce degree heterogeneity in spa- tial networks with total cost constraint. We find that degree heterogeneity affects the optimal exponent for the link length distribution. Specifically, the exponent shifts to a smaller value. Also, the average shortest-path length can further decrease due to degree heterogeneity. More- over, we study the effect of degree heterogeneity on the dynamics taking place on the spatial networks, and we observe the same shifting phenomenon of the optimal ex- ponent. To begin our analysis, we first briefly describe the orig- inal spatial network model introduced in [16]. Nodes are located in a d-dimensional regular square lattice, where each site i is connected with its 2d nearest neighbors. Pairs of sites ij are randomly chosen to receive long-range connections with probability proportional to r−α ij , where rij is the Manhattan distance between sites i and j (i.e., the number of co

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