1.204 Final Project Network Analysis of Urban Street Patterns

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1 1.204 Final Project Network Analysis of Urban Street Patterns Michael T. Chang December 21, Introduction In this project, I analyze road networks as a planar graph and use tools from network analysis to measure its structural properties. Network analysis provides a new way to quantify properties of road networks, which allows us to compare cities and regions with different road layouts. This is particularly useful in comparing different road layouts, which was the approach in [Cardillo et al., 2006]. In [Cardillo et al., 2006], road layouts across the world were divided into six categories. For example, New York City has a grid-iron layout, consisting of a planned grid of perpendicular streets. London has a medieval layout, consisting of randomly-oriented streets, resulting from unplanned growth, and is usually found in older cities. Irvine, California has a lollipop layout, consisting of many cul-de-sacs and dead ends, which is a layout often found in surburban areas. While it is easy to see that these layouts differ from one another, we will use network properties, like cost and efficiency, to quantify the difference. 2 Methods and data 2.1 Representing roads as a network There are two ways that a road network can represented as a mathematical graph. The first way is a primal representation, where intersections are nodes, streets are edges, and street lengths are weights, resulting in an undirected, weighted network. This representation is intuitive, because when visualized it resembles a map, as in Figure 1. It also the road s spatial information of locations and lengths. The alternative is a dual representation, where nodes are streets and edges are intersections. This representation discards the spatial info since long streets with many intersections are collapsed into singles nodes with many edges. This representation focuses on the connectivity of the streets and is more useful for studying certain properties, like betweenness centrality, as in [Crucitti et al., 2006]. An algorithm for creating a dual representation are described in [Masucci et al., 2009]. Choosing between a primal and dual representation depends on what properties of the road network are to be studied. For this project, I wanted to focus on the spatial and geographical properties, so the primal representation of a network was the better choice. 2.2 Data The two cities chosen for study in my project are San Francisco, CA, USA, and Oldenburg, Germany. San Francisco, being relatively modern, has large areas with a grid-iron layout, whereas 1

2 Figure 1: Primal representation of San Francisco s street network, containing 9322 nodes and edges. Oldenburg is a much older cities, with a largely medieval road layout. The primal representations of the road networks of these cities are shown in Figures 1 and 2. The data used for this project was obtained online. It was provided by the Computer Science department at Florida State University 1, as text files with spatial locations of nodes and connectivity of edges. They also have data on a number of other road networks, including other cities and highways in North America. There are some flaws in the data (e.g. intersections that don t actually exist), but these errors are small and will have a negligible impact on the network properties we will measure. 2.3 Network properties of interest In measuring network properties of road networks, I follow the same approach as in [Cardillo et al., 2006]. There are three structural properties that will be analyzed: the meshedness coefficient, the cost, and the efficiency. The meshedness coefficient M measures the degree of clustering. It is defined as M = F/F max, where F is the number of faces in the graph, and F max is the number of faces in the maximally connected graph. The formulas for these quantities, as given in [Cardillo et al., 2006], in a graph with N nodes and K edges are F = K N + 1 and F max = 2N 5. The meshedness coefficient is a replacement for the clustering coefficient that we are familiar with from class. The clustering coefficient is unsuitable for planar graphs, because it only counts triangles, whereas larger cycles 1 2

3 Figure 2: Primal representation of Oldenburg s street network, containing 6105 nodes and 7035 edges. 3

4 Figure 3: Demonstration of how network efficiency is calculated. The blue line represents the Euclidean distance between two nodes in the network, and the red line traces the shortest path between the points through the network. The efficiency for these two points is the ratio between the length of the blue and the red line. Global network efficiency is the average of this ratio, for all pairs of nodes. (squares, etc.) are important features of road networks. An illustration of this problem is that many different planar graphs all have clustering coefficient of 0, such as a square grid and a tree. The cost W is defined as the sum of the lengths of all edges in the network. It is related to the real cost of the road network, since the cost of building and maintaining roads increases with the amount of roads built. The cost is given by the formula: W = i,j a ij l ij (1) The efficiency E of a network represents the efficiency of flow within the network, namely how easily it is to get from one node to another. The local measure of efficiency, for a pair of nodes i and j, is the ratio of the Euclidean distance d Eucl ij to the distance along the shortest path through the network d ij. This is illustrated in Figure 3. The global efficiency of the network is the average value of efficiency for every pair of nodes, and is given by the formula: E = 1 N(N 1) i,j,i j d Eucl ij d ij (2) 2.4 Theoretical cases as baselines for comparison In other studies of networks, the random graph and the complete graph frequently serve as extreme cases and are useful for comparison. However, these theoretical models not appropriate for planar graphs, because the networks have intersecting edges. Instead, [Cardillo et al., 2006] proposes new theoretical models to represent the minimally-connected and maximally-connected cases. 4

5 Figure 4: Illustration of theoretical cases used for comparison. Top-left: Map of a part of Savannah, GA. Top-right: network representation of roads. Bottom-left: minimum-spanning tree of the network. Bottom-right: greedy triangulation of the network. Image reproduced from [Cardillo et al., 2006] The minimal case with the fewest number of edges is the minimally-spanning tree (MST). This is a tree uses the minimum number of edges to ensure that all nodes are connected in one component, and in a way that minimizes the total length of edges. The maximal case is the greedy triangulation (GT). This is a graph that creates a maximally-connected planar graph by drawing triangles between nodes wherever possible, but also minimizes the total length of edges. An example of these cases are depicted in Figure 4. The minimally-spanning trees for San Francisco and Oldenburg road networks used in this project are shown in Figures 5 and 6. Using these baseline cases, we define relative cost W rel and relative efficiency E rel. These are useful because it allows comparison across cities, since the values are normalized and account for differences in node layouts between cities. However, I didn t have time to write a greedy triangulation algorithm for the project, so only the MST is available for comparison. Consequently, I will be measuring cost of the network as a multiple of the cost of the MST, and measuring simply the absolute values of efficiency. W rel = W W MST W GT W MST (3) E rel = E EMST E GT E MST (4) 5

6 Figure 5: Minimum-spanning tree for San Francisco s street network Figure 6: Minimum-spanning tree for Oldenburg s street network. 6

7 Degree distribution San Francisco Oldenburg 0.5 Frequency, as a fraction Degree Figure 7: Degree distribution for street networks in San Francisco and Oldenburg. Notice that the average degree of Oldenburg s street network is lower, because its layout involves less 4-way intersections, which are usually only found in a grid-like layout. Compared to other networks, this degree distribution is very narrow, due to the planarity constraint. 3 Results 3.1 Degree distribution Figure 7 shows the distribution of node degree for both road networks. The very narrow range of degree is a result of the planarity constraint. The average degree is lower in Oldenburg than in San Francisco because its road layout uses 3-way intersections more than the 4-way ones found in grids. This is characteristic of an older street layout that was created in an unplanned, self-organized way. The results from [Cardillo et al., 2006] confirm this trend on a larger scale, where the findings indicate that, in general, P (k = 3) > P (k = 4) for self-organizaed street layouts, as in Cairo and Venice, but the reverse is true for planned cities, such as New York, San Francisco, and Washington. Figures 8 and 9 are maps of San Francisco and Oldenburg with intersections colored by node degree. The results are expected, and the map reinforces our expectation from what we see visually. Intersections in grid-like areas have degree 4, and intersections in areas with irregular street patterns (like the hilly area in SF, in Figure 8) have degree 3. Intersections in Oldenburg in general have lower degree overall. Interestingly, it appears that the nodes with higher degree are spaced randomly in SF, but are more concentrated in the city center for Oldenburg. This may reveal a tendency for roads to connect to existing intersections in self-organized growth, but not in pre-planned layouts. 7

8 Grid Degree 5900 Hills Figure 8: Map of node degree, for San Francisco s street network. Note the contrast in typical intersection degrees between the grid-like and hilly regions. Network Meshedness Cost Efficiency Coefficient (relative to MST) San Francisco, CA MST GT 1 Oldenburg, Germany MST GT 1 San Francisco, CA (Cardillo et. al) Irvine, CA (Cardillo et. al) London, UK (Cardillo et. al) Table 1: Network properties of road networks. The last three rows are taken from [Cardillo et al., 2006]. 8

9 Degree Figure 9: Map of node degree, for Oldenburg s street network. 9

10 3.2 Structural properties Table 1 compares the three properties of interest for various cities. Oldenburg has a much lower meshedness coefficient than San Francisco or any other grid-like city. This is expected, because grid-like cities have intersections that are more clustered, whereas medieval layouts may have intersections near each other but don t have a road between then. Irvine, CA also has a low meshedness coefficient, due to the disconnectedness of its roads. Low meshedness coefficient also reflects the relatively fewer number of streets compared to intersections in these cities. The road network in San Francisco costs 2.29 times more than its MST, but Oldenburg costs only 1.3 times more. The Oldenburg road network is more minimalist, containing fewer redundant connections, but this may increase difficulty in navigation. This may be a result of self-organized growth, because people may prefer to build streets only when absolutely necessary to connect two points. This difference in cost could be predicted from noticing that more streets are lost in building the MST for San Francisco than for Oldenburg. San Francisco s road network has a slightly higher efficiency than Oldenburg, and both have much higher than either MST, and Irvine. Interestingly, San Francisco s planned road network performs similarly to Oldenburg s self-organized network in terms of efficiency, despite having a higher cost. Also, notice that Irivine has a similar efficiency to the MSTs, which is expected, since the structure of the lollipop layout resembles an MST, consisting of a tree-like structure with many dead-ends. This result highlights the inefficiencies associated with such a road layout. 4 Conclusion These results can be useful in trying to design a transportation-effective and cost-effective road network. While in this project I only looked at the cost and efficiency of two cities, [Cardillo et al., 2006] compared 20 cities, and the results are shown in Figure 10. This graph plots each city on a graph of efficiency vs. cost. First, there seems to be an upper limit to efficiency. Secondly, we see that efficiency increases with cost, up to a certain point. It seems that the grid-iron layouts cost more than medieval layouts without achieving higher efficiency. Finally, the trend in this graph may imply an intrinsic trade-off between efficiency and cost independent of road layout. However, it is important to note that these metrics don t account other important properties of road networks, such as difficulty of navigation or amount of information required to locate an address. Further steps for this project would be to perform the analysis on larger-scale road networks (i.e. highways) to see if similar trends can be observed. I hypothesize that the network properties (meshedness coefficient, cost, and efficiency) would be different from those of local roads because highways are designed with different rules and goals. Future work could also account for other properties of road networks, such as betweenness centrality, traffic flows, or susceptibility to congestion. One possible question would be, how many roads can be removed before the network fails catastrophically and leads to congestion? References Alessio Cardillo, Salvatore Scellato, Vito Latora, and Sergio Porta. Structural properties of planar graphs of urban street patterns. Physical Review E, 73(6):066107, June ISSN doi: /PhysRevE URL

11 Figure 10: Efficiency vs. cost for various real-world road networks. Paolo Crucitti, Vito Latora, and Sergio Porta. Centrality measures in spatial networks of urban streets. Physical Review E, 73(3):036125, March ISSN doi: /PhysRevE URL a. P. Masucci, D. Smith, A. Crooks, and M. Batty. Random planar graphs and the London street network. The European Physical Journal B, 71(2): , August ISSN doi: /epjb/e URL epjb/e