Eur. Phys. J. B (2013) 86: 74 DOI: 10.1140/epjb/e2012-30235-7
THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Exploring the patterns and evolution of self-organized urban street networks through modeling Yikang Rui1,a , Yifang Ban1 , Jiechen Wang2 , and Jan Haas1 1 2
Geoinformatics, Royal Institute of Technology, 10044 Stockholm, Sweden Department of Geographic Information Science, Nanjing University, Nanjing 210093, P.R. China Received 15 March 2012 / Received in final form 22 July 2012 c EDP Sciences, Societ` Published online 1st March 2013 – a Italiana di Fisica, Springer-Verlag 2013 Abstract. As one of the most important subsystems in cities, urban street networks have recently been well studied by using the approach of complex networks. This paper proposes a growing model for selforganized urban street networks. The model involves a competition among new centers with different values of attraction radius and a local optimal principle of both geometrical and topological factors. We find that with the model growth, the local optimization in the connection process and appropriate probability for the loop construction well reflect the evolution strategy in real-world cities. Moreover, different values of attraction radius in centers competition process lead to morphological change in patterns including urban network, polycentric and monocentric structures. The model succeeds in reproducing a large diversity of road network patterns by varying parameters. The similarity between the properties of our model and empirical results implies that a simple universal growth mechanism exists in self-organized cities.
1 Introduction Transportation and infrastructure networks play crucial roles in the natural and social world [1], including plant leaves [2], mammalian metabolic and circulatory systems [3,4], rivers [5,6], airline networks [7–9], public transport networks [10,11] and internet [12,13]. Most of these networks are embedded in a d-dimensional Euclidean space which implies that the Euclidean coordinates of nodes and the physical lengths of edges are meaningful. Thus, some of the previous studies focused on the spatially preferential attachment mechanism in network evolution [14,15]. With optimization involved, Barth´elemy and Flammini [16] proposed a cost function of both the metric length and idealized traffic passing through the links. They obtained an optimal traffic tree with two extreme classes: minimum spanning tree and shortest path tree. Xie et al. [17] presented a growing network model with an optimal policy involving topological degree and geographical distance measures. Among diverse transport networks, we concentrate on the urban street network, which is one of the most important subsystems determining urban form and structure. Despite the complicate geographical, social and economical effects on street network evolution, recent empirical results have shown that there are some quantitative similarities for different street patterns. Supposing an undirected graph G = (V, E) is adopted with nodes V representing road intersections or end-nodes (cul-de-sacs) and a
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edges E representing road segments, we obtain a primary representation of a road network. Since edges do not cross each other, the graph G is called a planar graph [18]. Random planar graph models are naturally considered to roughly represent road networks or cities [19]. Fractal models are also introduced to study scale invariance of road networks [20]. Masucci et al. [21] pointed out that a growing model however shows more articulated properties than static counterparts, after comparing properties of London street network with three different models, i.e. a growing random planar graph, a static Erdos-Renyi random planar graph and a regular grid. From the perspective of complex networks, Barth´elemy and Flammini [22,23] proposed a model of urban street patterns with the shortest road-length of new road construction and loops generation. This model reproduces several empirical findings. Taking geometrical aspects and more parameters into consideration, Courtat et al. [24] recently presented a global mathematical model based on the division and extension of the space principle. In order to further explore the effects of the location choice of new centers and connection strategy on the evolution of self-organized urban street networks, we build a growing model with two characteristics: (1) new centers competitions with different values of attraction radius generate diverse patterns to explore growth mechanism of the self-organized road networks; (2) local optimization of connections and loop constructions are investigated to examine their effects on structure and topological properties. In Section 2, a road growth model is built and
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described in details. Section 3 investigates the local attachment optimization and competition process among new nodes by analyzing topological measures between different patterns and empirical results. Finally, Section 4 draws the conclusions.
2 Road growth model Transport systems are the veins and arteries of urban areas by moving people and goods between multiple origins and destinations to support economic and social activities, which eventually form complex spatial structures as cities [25,26]. As one important part of a transport system, the road network reflects the distribution of traffic flow [27] and its enlargement could approximate the urban growth. In microscopic view, the driving factors of urban growth include, among others, interactive individuals, economical behaviors and land development. From a macroscopic perspective, urban growth can lead to either urban sprawl or a compact and smart city. In addition, cities grow either in a self-organized process over time with an organic road network pattern (i.e. out of the control of any central agency) or in a single plan with a regular grid-like structure. When we disassemble road networks at a fine level, more typical patterns are identified, such as linear, treelike, radial, cellular and hybrid [28]. 2.1 General model description The following part will present a simple growing urban street network model with a local optimal policy. The model starts with a given node located in the center of the simulation area. During each time step t, N nodes are generated and only one of them with the maximum or minimum utility value in the competition process will be confirmed as a new center i. The center node will optimally connect to the current road network instead of intersections and end-nodes. The following is a list of some important definitions for nodes. – Characteristic nodes (Vc ): intersections and end-nodes in the primary representation of a road network [29]. We ignore the junctions with degree 2 in analysis in consist with the analysis in empirical studies. – Virtual nodes (Vv ): there are numerous virtual nodes uniformly distributed over each road segment. They are not involved in the analysis until they are connected to a new center and thus are converted into intersection nodes. – Existing nodes (Ve ): characteristic and virtual nodes. – Visible nodes (Vi ): existing nodes which are visible to the center i to ensure that there is no edge crossing. 2.2 Competition of new centers New center nodes, which may represent new homes, businesses, factories, etc., are normally dependent on existing road network in a real city. Location choice of a new
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center relates to numerous spatial, social, economic and environmental factors. Many location choice models have been developed in urban micro-simulations [30–32]. This presented approach simply focuses on nodes density and degree. High nodes density usually means high population and more demanding, while nodes with high degree imply they are more important from a topological perspective. We generate N (N > 1) new nodes with random places. For each new generated node i , we calculate its utility value U (r) of location choice: kj , (1) U (r) = j ∈Vc
where j is one node in Vc , which is the subset of Vc within radius r of the node i . kj is the degree of node j . If we minimize the value of U , new places are far away from current built-up area and eventually a uniform density of roads is obtained. Meanwhile, selection of the maximum U value generates interesting center distributions according to the attraction radius r. 2.3 Connection process In the connection process, we first compute the visible nodes set Vi for one center node i. For each node j in Vi , we then calculate: Sj = dij /kjα , (j ∈ Vi ),
(2)
where dij is the geographical distance and kj is the degree of node j. The center node i connects the node j with minimal Sj value, i.e. nodes with a large degree and short distance will be connected with the new node. If α = 0, only distance decides the choice. The optimality principle is implemented locally in both space and time during the growth, which well reflects planning limitation in the whole evolution histories of road networks in the real world. Note that any new connection should meet the angle test, i.e. the angle must be larger than a threshold Ac to prevent small-angle situation, which is not practical for real-world road networks. 2.4 Loop construction The growth mechanism so far leads to a tree-like structure. The next step is a loop construction process with the construction probability Pl , Pl ∈ [0, 1]. For larger Pl , more loops are generated with two methods. The first one is the relative neighborhood [2,22]. For the center node i and one visible node j, j ∈ Vi , suppose dij is defined as geographical distance between two nodes. Two circles with radius dij are drawn around node i and j, respectively. Node j is in the relative neighborhood of i only if the intersection of two circles contains no other existing nodes. The second method is less strict and creates rectangles with two constraints: dij can not be too longer than the segment generated in the connection process; the angle is around π/2.
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3 Simulations and discussions The model is implemented in an 1×1 unit square. For each time step t, one center is selected from the competition of N new centers. With incrementing t, more centers involve and road density increases. If N equals 1, there is no more competition and new centers are located randomly. N is an important parameter, however the influence of N on network growth would not be discussed in this paper. In our simulation, we set N ∼ tλ and choose a small λ. In contrast to previous models [22–24], parameters α and Pl are investigated to recover the influence of the connection and loop construction processes on the performance of the entire network. Furthermore, attraction radius r is also an interesting parameter when we select a new center, which has no predefined spatial distribution. Altering r leads to morphological changes from a urban network, a polycentric city to a monocentric city. Properties of diverse developed structures are investigated and compared with empirical studies to understand the mechanism of evolution in real cities.
3.1 Local optimization Many optimal network models involve two antagonistic quantities [1], e.g. total geometrical length and average shortest path [33]. Gastner and Newman [34,35] showed that their model displays striking similarities from roads to airline networks. In road network modeling, the following requirements are met: minimizes the total geometrical length subject to the average number of hops. The differences between our model and normal optimal network models are: (1) we build a growing model instead of a static one with given nodes, i.e. we use a local optimal policy instead of a global optimization; (2) the road network is represented as a planar graph, thus edge crossing is not allowed. In the following paragraphs we investigate the parameters α and Pl in connection and loop construction processes. If α = 0, only distance is under consideration. With increasing α, the new center is more likely to connect more important nodes (high degree) and thus the average degree of the entire network increases. Apparently there is an upper limit for α, since the angle of two adjacent roads cannot fall below a certain threshold in real-world road networks. The parameter Pl is the loop construction probability for the road network modeling. If Pl = 0, there is only one tree structure generated.
3.1.1 Organic and meshedness coefficient To measure whether a street network is organic or planned, we use the quantity: organic ratio [24], which is defined as follows: N (1) + N (3) , rN = j=2 N (j)
(3)
Fig. 1. (Color online) Properties of road networks generated with different α = 0, 0.4, 0.8, and Pl = 0.2 (black square), 0.6 (red circle) and 1 (blue upward triangles). All results have been obtained by averaging over 30 independent simulations.
where N (j) indicates the number of nodes with degree k = j. Since self-organized cities have N (3) > N (4), and single-planned cities most likely have square-like grid structures, rN 1 if the street network is organic and rN 0 if it is well planned. To measure whether a street network is tree-like structure or not, we use the quantity meshedness coefficient [36], which is originally named “alpha index” [37,38]. It is defined as the ratio of the existing and the maximum number of faces: M = (e − v + 1)/(2v − 5).
(4)
M equals to 0 if the network only consists of tree structures and is 1 if it is a complete planar graph. M is usually small because of the lack of triangles in real cities [36]. Besides the suburban tree structure, both organic selforganized and grid planned patterns can yield relatively large meshedness coefficients [39]. It can be observed that the meshedness coefficient M is an increasing function of both α and Pl , while the organic ratio rN shows an opposite trend (top panels of Fig. 1). Specifically, a small Pl value significantly reduces M and leads to a tree-like structure. Once a medium Pl value is chosen, the results fit well with empirical data (an average M of 20 real-world cities is 0.219) [39]. Moreover, if a large α is selected, more high degree nodes (k ≥ 4) emerge and cause a small rN , since rN measures the abundance of dead ends and T-shaped intersections (i.e. k = 1 and k = 3, respectively). Furthermore, with incrementing loop constructions, both the total topological length (LT ) and total geometrical length (LG ) increase naturally. However, with α increasing, the geometrical distance becomes less important and a new center connects to the existing node with higher degree even though the link is longer. Therefore, LT decreases and LG increases (bottom panels of Fig. 1), i.e. although new connections do not have the shortest geometrical lengths, they reduce the total number of hops
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and turns to other nodes. So the performance can be a balanced by optimizing the parameter α.
3.1.2 Efficiency Global efficiency Eglob is proposed by Latora and Marchiori [40], and measures how well the nodes communicate over the network. It is defined as: Eglob = (1/v(v − 1)) dij , dEucl (5) ij i,j∈V,i=j
is the Euclidean distance between nodes i where dEucl ij and j along a straight line. The shortest path dij can be measured as the minimum number of edges from node i to j in an unweighted network, where each edge is valued as one, or the smallest sum of the edges lengths throughout all possible paths in a weighted road network, where each edge is weighted by the geometrical length. Thus, we obtain two different measures of global efficiency, Eglob,T and Eglob,G , with topological and geometrical distance, respectively. Buhl et al. [36] showed that the geometric efficiency Eglob,G for street networks varies greatly from 0.4 to 0.837, with a majority of values lying between 0.7 and 0.8. Similar results are reported by Cardillo et al. [39], who pointed out that most cases in their empirical study achieve 80% of the maximum efficiency value. The topological global efficiency Eglob,T also shows much variability ranging from 0.104 to 0.287. Both quantities seem to decrease with an increase in network size in terms of total vertices. For increasing α, the global geometrical efficiency Eglob,G does not change too much, however the global topological efficiency Eglob,T grows impressively. Therefore, α is an important parameter for topological measures instead of geometrical ones. Conversely, Eglob,G is more sensitive to Pl than Eglob,T . When Pl reaches 0.6, Eglob,G is about 0.8, which can be found in most of the cases [39]. A medium value of Pl is also the best choice for Eglob,T . It implies that building too many loops cannot infinitely improve and even may worsen the global topological efficiency (Fig. 2).
3.2 Overall morphological changes If attraction radius r equals 0, a new center is chosen randomly from N new generated nodes and final center distribution is almost uniform. When r is small, center nodes are located dispersedly and are centralized in many small areas. With increasing r, nodes concentrate in one area. The value of r to be chosen in our analysis is examined by evaluating the following topological attributes: fraction of dominating sectors and Gini index. Suppose we divide our simulated space into square sectors with equal area. There are totally N c center nodes and N s sectors. N (i) is the number of center nodes in the ith sector. According to the distribution of nodes in
Fig. 2. (Color online) Global geometrical and topological efficiency for networks with increasing parameters: α = 0, 0.4, 0.8, and Pl = 0.2 (black square), 0.6 (red circle) and 1 (blue upward triangles). All results have been obtained by averaging over 30 independent simulations.
space, we compute the fraction of dominating sectors [23] as follows: σ=
N (i) 2 1 , Y2 = . Y2 N s Nc i
(6)
In a uniform case, all N (i) are almost the same and thus σ ∼ 1. However if most nodes concentrate in a few (n) sectors, we obtain Y2 ∼ 1/n and σ ∼ n/N s. The Gini index is adopted to measure the inequality of traffic distribution in a network [41,42]. Gini = 1 −
e k=1
(Xk − Xk−1 )(Yk − Yk−1 ),
(7)
where e is the number of edges. Xk and Yk represent the cumulative portion of links and total traffic flow, respectively. A value of 0 expresses total equality, while a value of 1 represents maximal inequality. We use edge betweenness centrality (CeB ) to represent ideal traffic flow. Similar to node betweenness centrality, which will be introduced in a later paragraph, CeB measures the fraction of the shortest paths between any couples of nodes that go through the edge e. Figure 3 shows the results of the fraction of dominating sectors σ and the Gini index for different values of r. We observe that σ decreases very fast with increasing r. When r is greater than 0.075, σ changes very slightly. There is a clear peak distribution for the Gini index. When r is around 0.025, the Gini index reaches its maximum, i.e. the maximum inequality status for idealized traffic flow. Figure 4 shows four typical patterns with different values of r: 0 (a), 0.025 (b), 0.075 (c) and 0.25 (d).
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(a)
(b)
(c)
(d)
Fig. 3. (Color online) Fraction of dominating sectors (top panel) and Gini index (bottom panel) of road network simulations with different attraction radius r (t = 1000, α = 0.4, Pl = 0.6 and averaged over 100 configurations).
3.2.1 Cellular structure A road network can be called as a two-dimensional cellular system by assuming that small land patches are enclosed by roads. L¨ammer et al. [41] studied the 20 largest German cities and found that the frequency distribution P (A) of the cells areas A obeys the power law P (A) ∼ A−τ with τ ranging from 1.74 to 2.16. A similar power-law distribution with slope –2 is found for the London street network by [21]. Simulations show that area distribution is very sensitive to centers (intersections or end-nodes) distribution [22]. After removing the tree structure, the P (A) for different values of r is measured. When r = 0, the nodes distribution is almost uniform and P (A) is exponential [23]. Figure 5 shows a power-law distribution P (A) for r = 0.25 with an exponent τ close to 2.5. Furthermore, with increasing r, the value of τ in the inset declines quickly from 4.6 (r = 0.0125) to 1.05 (r = 0.05), and then climbs to 2.5 (r = 0.25). The results imply that the obtained pattern when r = 0.05 is the most heterogeneous. It is interesting to check the empirical studies about the road network evolution presented by Strano et al. [43]. Over the last two centuries, the Groane area changed from a polycentric region (29 urban centers) into a completely urbanized area. Meanwhile τ changes over time from 1.2 to 1.9 in 1833 and 2007, respectively, because of the process of homogenization of the cells size. Our simulation results are consistent with this observation. The pattern for r = 0.05 displays a small τ and thus is suitable to model a polycentric region. 3.2.2 Betweenness centrality Centrality measures are crucial for understanding the structure properties of complex networks by quantifying some nodes are more important than others. Betweenness
Fig. 4. Road networks generated in each 1 × 1 unit square with t = 1000, α = 0.4, Pl = 0.6, and r = 0 (a), 0.025 (b), 0.075 (c), 0.25 (d), respectively. The red star in the middle represents the initial node.
centrality for vertex i, CiB , is based on the idea that a node is central if it is traversed by many of the shortest paths between couples of nodes [44]. It is defined as: CiB = (1/(v − 1)(v − 2))
njik /njk ,
(8)
j,k∈V, j=i=k
where njk is the number of shortest paths between node j and k, and njik is the number of shortest paths between node j and k that contain node i. Betweenness centrality distribution is peaked (exponential for self-organized cities and Gaussian for planned cities) [45,46]. However, the peaked distribution does not seem to be universal. L¨ ammer et al. found it very broad for German cities with a power-law exponent ranging from 1.279 to 1.486 [41]. Figure 6 shows the cumulative distributions of node betweenness centrality for different r values. It exhibits an exponential distribution P (B) ∼ exp(−B/s), with coefficient s ranging from 0.069 (r = 0.025), to 0.027 (r = 0.25). The pattern with r = 0.25 shows a symmetrical structure and the distribution of betweenness centrality does not vary too much. The pattern for r = 0.025 reveals that some nodes have extremely high betweenness values because they lie in crucial positions most paths pass through. Thus, a polycentric city is very likely to have a larger s value compared with monocentric one. In previous research [45], Venice has the largest exponent s = 0.044, with strong environmental and physical constraints. Our
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Fig. 5. (Color online) The area distribution for the pattern with r = 200 (t = 1000, α = 0.4 and Pl = 0.6) displays a power-law behavior P (A) ∼ A−τ . The inset shows the value of exponent τ for different r. All results are averaged over 100 configurations.
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(a)
(b)
(c)
(d)
Fig. 7. Simulations with varying r from 0.25 to 0.025 (top panels) and from 0.025 to 0.25 (bottom panels) at two growth status ((a) and (c) at t = 500; (b) and (d) at t = 1000) respectively (α = 0.4 and Pl = 0.6).
Fig. 6. (Color online) Cumulative distribution of betweenness centrality for different values of parameter r (t = 1000, α = 0.4 and Pl = 0.6). All results are averaged over 100 configurations.
experiments confirmed the observation by considering that environmental and physical constraints usually lead to fewer roads (bridges or tunnels) with extremely high traffic flow or even polycentric structures of cities. 3.2.3 Simulations with varying r Constant attraction radius r is not realistic for modeling a real city from the perspective of historical evolution. Here, varying values for parameter r are investigated. We test both decreasing and increasing values of varying r (Fig. 7). For decreasing r, we alter r from 0.25 (t = 0) to 0.075 (t = 200) and finally 0.025 (t = 600). Panels (a) and (b) are two growth status at t = 500 and 1000, respectively.
For increasing r, r changes from 0.025 (t = 0) to 0.075 (t = 200) and to 0.25 (t = 600). The pattern for decreasing r simulates how a city grows into an asymmetric structure with some emerging sub-centers. This pattern is quite common for self-organized cities due to environmental and physical constraints in our real world. Compared with the pattern for r = 0.25, which can be denoted as a relative symmetrical structure, the pattern in Figure 7b displays a larger Gini index (0.65) and larger exponent s of betweenness centrality distribution (0.036). In Figure 7d, starting with several towns, the pattern for increasing r changes towards one agglomerated urban area (a dominating city). This evolution is realistic in our world [43] and our model can be explained from a historical perspective. In olden times, people’s activities were limited in space and they normally got local information for location choices, due to spatial constraints, inconvenient transportation, etc., as a result a system of many small villages or towns began to form. With people’s ability to overcome the spatial constraints and acquire more information, new population tended to shift towards in big cities for better lives. For these central cities, further urban development strategies may be dispersed, compact or polycentric development [47]. Although our model does not show the phenomenon of urban sprawl, it provides two scenarios of multi-centred developing processes: sub-centers were originally small towns and later become absorbed by the main city’s spread (Fig. 7d); and subcenters arise at high accessible transport nodes (Fig. 7b).
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4 Conclusions Cities are complex systems and network theory is regarded as a powerful tool to study urban issues, especially urban transportation systems. Our objective is to model selforganized urban street networks and explore the patterns and evolution from the perspective of complex networks. As a planar graph in the primary representation, road networks normally show little properties of classical complex networks due to spatial constraints and the limitation of nodes degree. However, after fully reviewing the topological properties of road networks in real-world cities, we find that some quantitative similarities do exist for different road patterns. In this paper, we build a growing model for selforganized urban street networks. In the local optimal attachment process, adjusting the preferential parameter α results in changes in degree distribution and two antagonistic quantities: total geometrical length and total number of hops or turnings. If setting α appropriately, we could achieve a balance between construction costs and traveling efficiency. In the loop construction process, the topological properties of our model with a medium value of Pl match observations from the empirical studies. Particularly, our experiments show that building more loops cannot infinitely improve the global topological efficiency of a road network. Moreover, during the competition process for new centers with different attraction radius r, several typical patterns emerge including urban network, polycentric and monocentric structures. We investigate the properties of different patterns and observed that the power-law cell area distribution yields the smallest exponent when r is around 0.05. The distribution of betweenness centrality is exponential with a decreasing exponent when r is increasing. The changes of the exponent reflect the homogeneity of both cell area and idealized traffic flow for different patterns. Interestingly, by setting the varying r during network growth, we construct two different patterns with an asymmetric structure that well simulate the evolution of road network at both regional and urban levels. It thus implies that the varying parameter r is more realistic for modeling and can be a valuable asset in studying the coevolution of the entire network and the location of new centers. In summary, from local optimization in connection process to overall morphological changes with different values of attraction radius, we believe that our model can be an interesting approach to explore the growth mechanisms of diverse patterns of the self-organized urban street networks. The authors thank Professor Bin Jiang and two anonymous referees for constructive discussions and comments on this paper.
References 1. M. Barth´elemy, Phys. Rep. 499, 1 (2011) 2. A. Runions, A.M. Fuhrer, B. Lane, P. Federl, A.-G. Rolland-Lagan, P. Prusinkiewicz, ACM Trans. Graph. 24, 702 (2005)
Page 7 of 8 3. J.R. Banavar, J. Damuth, A. Maritan, A. Rinaldo, Nature 421, 713 (2003) 4. J.R. Banavar, M.E. Moses, J.H. Brown, J. Damuth, A. Rinaldo, R.M. Sibly, A. Maritan, Proc. Natl. Acad. Sci. USA 107, 15816 (2010) 5. I. Rodriguez-Iturbe, A. Rinaldo, Fractual River Basins: Chance and Self-Organization (Cambridge University Press, Cambridge, 1997) 6. J.R. Banavar, F. Colaiori, A. Flammini, A. Maritan, A. Rinaldo, J. Stat. Phys. 104, 1 (2001) 7. A. Barrat, M. Barth´elemy, R. Pastor-Satorras, A. Vespignani, Proc. Natl. Acad. Sci. USA 101, 3747 (2004) 8. R. Guimera, S. Mossa, A. Turtschi, L.A.N. Amaral, Proc. Natl. Acad. Sci. USA 102, 7794 (2005) 9. G. Bianconi, P. Pin, M. Marsili, Proc. Natl. Acad. Sci. USA 106, 11433 (2009) 10. J. Sienkiewicz, J.A. Holyst, Phys. Rev. E 72, 046127 (2005) 11. C. von Ferber, T. Holovatch, Y. Holovatch, V. Palchykov, Eur. Phys. J. B 68, 261 (2009) 12. E. Ravasz, A.-L. Barabasi, Phys. Rev. E 67, 026112 (2003) 13. R. Pastor-Satorras, A. Vespignani, Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge University Press, Cambridge, 2003) 14. C. Herrmann, M. Barth´elemy, P. Provero, Phys. Rev. E 68, 026128 (2003) 15. S.S. Manna, P. Sen, Phys. Rev. E 66, 066114 (2002) 16. M. Barth´elemy, A. Flammini, J. Stat. Mech. L07002 (2006) 17. Y. Xie, T. Zhou, W. Bai, G. Chen, W. Xiao, B. Wang, Phys. Rev. E 75, 036106 (2007) 18. R. Diestel, Graph Theory (Springer-Verlag, Heidelberg, New York, 2005) 19. S. Gerke, D. Schlatter, A. Steger, A. Taraz, Random Struct. Algorithms 32, 236 (2008) 20. V. Kalapala, V. Sanwalani, A. Clauset, C. Moore, Phys. Rev. E 73, 026130 (2006) 21. A.P. Masucci, D. Smith, A. Crooks, M. Batty, Eur. Phys. J. B 71, 259 (2009) 22. M. Barth´elemy, A. Flammini, Phys. Rev. Lett. 100, 138702 (2008) 23. M. Barth´elemy, A. Flammini, Netw. Spat. Econ. 9, 401 (2009) 24. T. Courtat, C. Gloaguen, S. Douady, Phys. Rev. E 83, 036106 (2011) 25. B.J.L. Berry, F.E. Horton, Geographic Perspective on Urban Systems (Prentice Hall, Englewood Cliffs NJ, 1970) 26. J.P. Rodrigue, C. Comtois, B. Slack, The Geography of Transport Systems (Routledge Education, Oxon, 2006) 27. B. Jiang, S. Zhao, J. Yin, J. Stat. Mech. P07008 (2008) 28. S. Marshall, Streets & Patterns (Spon Press, Taylor & Francis Group, New York, 2005) 29. B. Jiang, C. Claramunt, Transactions in GIS 6, 295 (2002) 30. J.E. Miller, D.J. Hunt, J.E. Abraham, P.A. Salvini, Comput. Environ. Urban Syst. 28, 9 (2004) 31. P. Wagner, M. Wegener, Urban Land Use, DISP 170, 45 (2007) 32. X. Li, X. Liu, Int. J. Geogr. Inf. Sci. 22, 21 (2008) 33. D.J. Aldous, Spatial Transportation Networks with Transfer Costs: Asymptotic Optimality of Hub and Spoke Models, in Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press, Cambridge, 2008), pp. 471–487
Page 8 of 8 34. M.T. Gastner, M.E.J. Newman, Eur. Phys. J. B 49, 247 (2006) 35. M.T. Gastner, M.E.J. Newman, Phys. Rev. E 74, 016117 (2006) 36. J. Buhl, J. Gautrais, N. Reeves, R.V. Sole, S. Valverde, P. Kuntz, G. Theraulaz, Eur. Phys. J. B 49, 513 (2006) 37. W.L. Garrison, D. Marble, US Army Transp. Command Tech. Rep. 62, 73 (1962) 38. K.J. Kansky, Structure of Transportation Network: Relationships Between Network Geometry and Regional Characteristics (University of Chicago Press, Chicago, 1963) 39. A. Cardillo, S. Scellato, V. Latora, S. Porta, Phys. Rev. E 73, 066107 (2006)
Eur. Phys. J. B (2013) 86: 74 40. V. Latora, M. Marchiori, Phys. Rev. Lett. 87, 198701 (2001) 41. S. L¨ ammer, B. Gehlsen, D. Helbing, Physica A 363, 89 (2006) 42. F. Xie, D. Levinson, Comput. Environ. Urban Syst. 33, 211 (2009) 43. E. Strano, V. Nicosia, V. Latora, S. Porta, M. Barthelemy, Sci. Rep. 2, 296 (2012) 44. L.C. Freeman, Soc. Netw. 1, 215 (1979) 45. P. Crucitti, V. Latora, S. Porta, Chaos 16, 015113 (2006) 46. S. Porta, P. Crucitti, V. Latora, Physica A 369, 853 (2006) 47. H. Salas-Olmedo, working paper 1034, TSU, Oxon. UK (2008)
