As described in sections 2.2.1, empirical studies in network science have studied the structural properties of real-world networked systems using measures such as those identified in section 2.2.2. In many of these studies, the main focus has been on whether a network displays small- world or scale-free properties against the null hypothesis that the network is random. These studies are the subject of section 2.2.4.1. It is also possible to identify a class of empirical studies that focus specifically on spatially constrained networks. Findings for these studies are the subject of section 2.2.4.2.
2.2.4.1 Detection of Small-World and Scale-Free Signatures
The method used to identify a small-world signature in a network compares calculated values of average shortest path length (calculated as the number of links)〈݈〉 and average clustering coefficient ܥ, with values for the same measures, denoted 〈݈〉ௗand ܥௗ, calculated on a
random counterpart graph; that is, a random network that has the same number of nodes and links (Watts and Strogatz, 1998). A small-world signature is said to have been detected if
〈݈〉 ≈ 〈݈〉ௗ and ܥ ب ܥௗ, i.e. if the average shortest path length of the network is
approximately similar to a random graph but it exhibits much greater clustering between nodes. After defining the small-world model, Watts and Strogatz (1998) went on to demonstrate evidence for the existence of small-world signatures in the neural network of a nematode worm, the power grid in the USA and a network of collaborations between actors in films. This evidence is shown in Figure 2.7 (ܮ ൌ 〈݈〉 in this figure).
Figure 2.7 - Evidence for small-worlds in the film actors network, US power grid and neural network of a nematode worm (C. elegans). (Table 1, Watts and Strogatz (1998))
The typical method used to identify whether a network has a scale-free signature is the production of a plot of node degrees ݇ against probability (݇) on a doubly logarithmic scale and to observe whether a straight line pattern emerges in the data points. This approach has been used, for example, by Faloutsos et al. (1999) and Albert et al. (1999) for the Internet and by Redner (1998) for two networks of citations between scientific papers. The plots produced by Redner (1998) are shown in Figure 2.8 as examples.
Figure 2.8 - Evidence of power law scaling in the degree distribution of two networks of citations between scientific papers. (Figure 1a, Redner (1998))
The above cited examples provide a small illustration of how empirical studies in network science have uncovered that many real-world network systems share similar structural characteristics. The survey papers of Albert and Barabasi (2002) and Costa et al. (2011)
summarise many more examples of the application of this approach and include results on the structural properties of a diverse range of systems including the Internet and World Wide Web, Biological networks, Social networks and Infrastructure networks. However, in recent years, there has also been criticism of these findings from other research disciplines. For example, statisticians have criticised the method of producing a log-log plot to detect a power law as overly simplistic at best or misleading and erroneous at worst, with some research finding that power laws do not exist where they were previously claimed to exist by others (Clauset et al., 2009). The most prominent example of this criticism is a claim that the Internet does not actually have a power law node degree distribution; for example, by Doyle et al. (2005). There has also been criticism that, in focusing primarily on the structural properties of the graphs that underlie networked systems, studies in network science neglect other important domain relevant information, which significantly limits the usefulness of their findings (Alderson, 2008, Havlin et al., 2012). For example, all of the studies cited in this section thus far focussed solely on the connectivity properties of networks, and omitted other characteristics of nodes and links. Whilst more recent empirical studies, which are described in the next section, have begun to include some of these other characteristics, primarily in the form of link lengths, there is still some way to go on this issue. As will be shown in sections 2.3 and 2.4, this criticism can also be made of studies of road traffic networks.
2.2.4.2 The Structural Properties of Spatially-Constrained Networks
In some of the networks discussed in the previous section; such as the citation network and network of actor collaborations, the difference in cost between connections of differing lengths is relatively small. In contrast, in networks such as the Internet and transportation networks, this is not the case because their geographical embedding imposes constraints on their formation and operational characteristics. This typically manifests in increased costs of long distance connections, which therefore require a strong economic reason for their construction (Barthelemy, 2011). These differing constraints make such networks an interesting class to study within the broader family of networks.
Barthelemy (2011) provides a review of empirical studies of these networks, which include studies of transportation networks such as:
Airline networks – This includes work by Barrat et al. (2004) on the worldwide air- transportation network and Barrat et al. (2005) on the link between airport size and the magnitude of travel. These studies found that the “airport connection graph is … a clear example of a spatial (non-planar) small-world network displaying a heavy-tailed degree distribution and heterogeneous topological properties”
Bus and Subway (Underground) networks – This includes work by Sienkiewicz and Holyst (2005) and von Ferber et al. (2009) on the public transport networks of 22 polish and 15 worldwide cities respectively. In comparison with airline networks, these networks appear to have smaller average node degrees and longer average shortest path lengths. Barthelemy (2011) hypothesised that this is a consequence of the more restrictive spatial constraints that exist in these networks.
Cargo ship networks – This includes work by Hu and Zhu (2009) and Kaluza et al. (2010) on the worldwide cargo ship network. In contrast to bus networks, Barthelemy (2011) hypothesised that these works appear to show that such networks are less constrained by their spatial embedding and that long distance links are actually less costly than short distance links in such networks.
In his conclusions, Barthelemy (2011) highlighted that spatial networks can be broadly split into categories: planar networks, such as bus and subway networks; and spatial, non-planar networks, such as airline and cargo ship networks, where nodes have a geographical embedding but where links can intersect. Networks in the latter category appear to have more similarities in structure to the networks studied in the empircal studies described in section 2.2.4.1. As will be shown in section 2.3, road traffic networks fall into the former category. Barthelemy (2011) also identified several important influences of spatial constraints on the structural properties of networks. Firstly, that they restrict the occurence of high degree nodes and usually produce a degree distribution that is peaked around the average degree rather than a power law like many other real-world networked systems; secondly, that spatial constraints limit the length of links and, for planar networks, the link length distribution is usually peaked; and thirdly, that restrictions on node degree in planar networks constrain the formation of hub nodes in favour of short links, which tends to lead to highly clustered networks.
Like the studies of the previous section, the studies highlighted here have been criticised for stripping out domain relevant information. For example, Derrible and Kennedy (2011) criticised the studies of bus networks because they did not include the fact that bus networks are composed of transit lines and that the ability to transfer between transit lines is not explicitly recognised.
2.2.5 Studies of the Effects of Structure on the Performance Characteristics of Networks