arXiv:cond-mat/0409702v1 [cond-mat.dis-nn] 27 Sep 2004 Shape and efficiency in spatial distribution networks Michael T. Gastner and M. E. J. Newman Department of Physics, University of Michigan, Ann Arbor, MI 48109 We study spatial networks that are designed to distribute or collect a commodity, such as gas pipelines or train tracks. We focus on the cost of a network, as represented by the total length of all its edges, and its efficiency in terms of the directness of routes from point to point. Using data for several real-world examples, we find that distribution networks appear remarkably close to optimal where both these properties are concerned. We propose two models of network growth that offer explanations of how this situation might arise. A network is a set of points or vertices joined together in pairs by lines or edges. Networks provide a useful framework for the representation and modeling of many physical, biological, and social systems, and have received a substantial amount of attention in the recent physics literature [1, 2, 3]. In this paper we study networks in which the vertices occupy particular positions in geo- metric space. Not all networks have this property���web pages on the world wide web, for example, do not live in any particular geometric space���but many others do. Ex- amples include transportation networks, communication networks, and power grids. Recently several studies have appeared in the physics literature that address the ways in which geography influences networks [4, 5, 6, 7, 8, 9]. In this paper we study the spatial layout of man-made distribution or collection networks, such as oil and gas pipelines, sewage systems, and train or air routes. The vertices in these networks represent, for instance, house- holds, businesses, or train stations and the edges repre- sent pipes or tracks. In most cases the network also has a ���root node���, a vertex that acts as a source or sink of the commodity distributed���a sewage treatment plant, for example, or a central train station. Geography clearly affects the efficiency of these net- works. A ���good��� distribution network as we will consider it in this paper has two definitive properties. First, the network should be efficient in the sense that the paths from each vertex to the root vertex are relatively short. That is, the sum of the lengths of the edges along the shortest path through the network should be not much longer than the ���crow flies��� distance between the same two vertices: if a subway track runs all around the city before getting you to the central train station, the train is probably not of much use to you. Second, the sum of the lengths of all edges in the network should be low so that the network is economical to build and maintain. In this paper we argue that these two criteria are often at odds with one another, but that even so, real networks manage to find solutions to the distribution problem that come remarkably close to being optimal in both senses. We suggest possible explanations for this observation in the form of two growth models for geographic networks that generate networks of comparable efficiency to our real-world examples. We begin our study by looking at the properties of some real-world distribution networks. We consider four examples as follows. Our first network is the sewer system for the City of Bellingham, Washington. From GIS data for the city we extracted the shapes and positions of the parcels of land (roughly households) into which the city is divided and the lines along which sewers run. We constructed a network by assigning one vertex to each parcel whose centroid was less than 100 meters from a sewer. The vertex was placed on the sewer at the point closest to the corresponding centroid and adjacent vertices along the sewers were connected by edges. The city���s sewage treatment plant was used as the root vertex, for a total of 23 922 vertices including the root. Our next two examples are networks of natural gas pipelines, the first in Western Australia (WA) and the second in the southeastern part of the US state of Illinois (IL) [16]. We assigned one vertex to each city, town, or power station within 10km (WA) or 10,000 feet (IL) of a pipeline. The vertex was placed on the pipeline at the point closest to each such place, and adjacent vertices joined by edges. The root for WA was chosen to be the shore point of the pipeline leading to the Barrow Island oil fields and for IL to be the confluence of two major trunk lines near the town of Hammond, IL. The resulting networks have 226 (WA) and 490 (IL) vertices including the roots. For our last example we take the commuter rail sys- tem operated by the Massachusetts Bay Transportation Authority in the city of Boston, MA (Fig. 1a). In this network, the 125 stations form the vertices and the tracks form the edges. In principle, there are two components to this network, one connected to Boston���s North Station and the other to South Station, with no connection be- tween the two. Since these two stations are only about one mile apart, however, we have, to simplify calcula- tions, added an extra edge between the North and South Stations, joining the two halves of the network into a single component. The root node was placed halfway between the two stations for a total of 126 vertices in all. We wish to quantify the efficiency of these networks in terms of path lengths and combined edge length, as de- scribed above. To do this, we compare our measurements

2 FIG. 1: (a) Commuter rail network in the Boston area. The arrow marks the assumed root of the network. (b) Star graph. (c) Minimum spanning tree. (d) The model of Eq. (3) applied to the same set of stations. of the networks to two theoretical models that are each optimal by one of these two criteria. If one is interested solely in short, efficient paths to the root vertex then the optimal network is the ���star graph,��� in which every ver- tex is connected directly to the root by a single straight edge (see Fig. 1b). Conversely, if one is interested solely in minimizing total edge length, then the optimal net- work is the minimum spanning tree (MST) (see Fig. 1c). (Given a set of n vertices at specified points on a flat plane, the MST is the set of n - 1 edges joining them such that all vertices belong to a single component and the sum of the lengths of the edges is minimized [17].) To make the comparison with the star graph, we con- sider the distance from each non-root vertex to the root first along the edges of the network and second along a simple Euclidean straight line, and calculate the mean ratio of these two distances over all such vertices. Fol- lowing Ref. [10], we refer to this quantity as the network���s route factor, and denote it q: q = 1 n n summationdisplay i=1 li0 di0 , (1) where li0 is the distance along the edges of the network from vertex i to the root (which has label 0), and di0 is the direct Euclidean distance. If there is more than one path through the network to the root, we take the shortest one. Thus, for example, q = 2 would imply that on average the shortest path from a vertex to the root through the network is twice as long as a direct straight- line connection. The smallest possible value of the route factor is 1, which is achieved by the star graph. The route factors for our four networks are shown in Table I. As we can see, the networks are remarkably efficient in this sense, with route factors quite close to 1. Values range from q = 1.13 for the Western Australian gas pipelines to q = 1.59 for the sewer system. We also show in Table I the total edge lengths for each of our networks, along with the edge lengths for the MST on the same set of vertices and, as the table shows, we again find that our real-world networks are competitive route factor edge length (km) network n actual MST actual MST star sewer system 23 922 1.59 2.93 498 421 102 998 gas (WA) 226 1.13 1.82 5 578 4 374 245 034 gas (IL) 490 1.48 2.42 6 547 4 009 59 595 rail 126 1.14 1.61 559 499 3 272 TABLE I: Number of vertices n, route factor q, and total edge length for each of the networks described in the text, along with the equivalent results for the star graphs and minimum spanning trees on the same vertices. (Note that the route factor for the star graph is always 1 and so has been omitted from the table.) with the optimal model, the combined edge lengths of the real networks ranging from 1.12 to 1.63 times those of the corresponding MSTs. But now consider the remaining two columns in the table, which give the route factors for the MSTs and the total edge lengths for the star graphs. As the table shows, these figures are for all networks much poorer than the optimal case and, more importantly, much poorer than the real-world networks too. Thus, although the MST is optimal in terms of total edge length it is very poor in terms of route factor and the reverse is true for the star graph. Neither of these model networks would be a good general solution to the problem of building an ef- ficient and economical distribution network. Real-world networks, on the other hand, appear to find a remarkably good compromise between the two extremes, possessing simultaneously the benefits of both the star graph and the minimum spanning tree, without any of the flaws. In the remainder of the paper we consider mechanisms by which this might occur. The networks we are dealing with are not, by and large, designed from the outset for global optimality (or near- optimality) of either their total edge length or their route factors. Instead, they form by growing outward from the root, as the population they serve swells and infrastruc- ture is extended and improved. To explore the possi- bilities of this process we consider a situation in which