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Complex Network Analysis on Transportation Networks
Emmanouil Chaniotakis1, Evangellos Mitsakis1, Josep Maria Salanova Grau1, Iraklis Stamos1
11 Εθνικό Κέντρο Έρευνας και Τεχνολογικής Ανάπτυξης, Ινστιτούτο Βιώσιμης Κινητικότητας και Δικτύων Μεταφορών
E-mail: [email protected], [email protected], [email protected], [email protected]
Περίληψη Το παρόν άρθρο παρουσιάζει την εφαρμογή της ανάλυσης πολύπλοκων δικτύων και της θεωρίας γράφου για τα συγκοινωνιακά δίκτυα. Ως αναφορά στην δομή τους και τις ιδιότητες τους τα δίκτυα που χρησιμοποιήθηκαν εξετάστηκαν σχετικά με την δυνατότητα κατηγοριοποίησης τους ως δίκτυα μικρόκοσμου (small world) και σχετικά με την ιδιότητα τους οι διαστάσεις τους να είναι ανεξάρτητες από την κλίμακα (scale-free). Χρησιμοποιήθηκαν δύο οδικά δίκτυα διαφορετικής ιεράρχησης και ένα δίκτυο αστικών συγκοινωνιών. Βρέθηκε ότι η δομή μικρόκοσμου μπορεί να αποδοθεί σε συγκοινωνιακά δίκτυα και στην Ελλάδα, ενώ κάτι τέτοιο δεν φαίνεται να ισχύει με την υπάρχουσα έρευνα για την ιδιότητα οι διαστάσεις να είναι ανεξάρτητες από την κλίμακα. Τα δίκτυα που εξετάστηκαν είναι το δίκτυο αυτοκινητοδρόμων της Ευρώπης, το οδικό δίκτυο της Θεσσαλονίκης και το δίκτυο αστικών μεταφορών της Θεσσαλονίκης. Για τα δύο τελευταία εφαρμόστηκαν οι δείκτες από της θεωρία γράφου και τα αποτελέσματά τους παρουσιάζονται και αναλύονται. Λέξεις κλειδιά: ΠολύπλοκαΔίκτυα, ΔίκτυαΜικρόκοσμου, ΔίκτυαΑνεξάρτηταΚλίμακας, ΘεωρείαΓράφου. Abstract
This paper presents the application of complex network analysis and graph theory measures for transportation networks. Concerning complex network analysis, the networks used were examined on their structure and their properties. The analysis focus on small world networks and the scale-free property. It was found that small world properties can be an appropriate modelling structure for transportation networks also in Greece while the scale-free property is not widely met. The analysis was performed on the highway network of Europe, the Thessaloniki urban network and the Thessaloniki Public Transport network. For the two latter, graph measures were applied and analysed.
Keywords:Complex Network Analysis, Small World Networks, Scale-Free, Graph Theory.
1. Introduction
Transportation infrastructure is a crucial component of economic growth and development and one of the highest both internal and external costs. Thus, the representation and analysis of transportation infrastructure is a scientific field that has received attention from the scientific community during the past decades. Commonly, transportation infrastructure is represented as a network; defined by a set of nodes connected by a set of links. The definition of Transportation Networks (TN) have allowed the representation and analysis of the infrastructure on the network properties. On a macroscopic scale, TN are usually analysed on the concept of network equilibrium and measures such as traffic flows, travel times and
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average speeds are used as performance indicators in order to evaluate the network structure and the impact of potential network alterations (Cascetta, 2009).
Lately, the development of Complex Networks Analysis on the premises of Graph Theory allowed for advances on the identification of network characteristics and alternative sets of measures for the evaluation of performance (Dorogovtsev & Mendes, 2002; Rodrigue et al., 2013). The introduction to the network graph characterization was presented in the seminal paper of Erdős and Rényi (1959) where random networks are defined. This definition allowed for advances in the premises of graph network modelling towards the definition of other types of networks that can exist and exhibit certain characteristics that are neither (a) regular networks ( 0) nor random networks ( 1) but fall in between those categories (0
1) based on the probability of connectivity randomness ( ). Two network types have received large attention due to modelling potential and the fact that many real world networks fall into this category (Amaral et al., 2000; Dorogovtsev & Mendes, 2002): the scale-free networks and the small world networks.
The main properties of small wold networks are (a) the preservation of the local neighbourhood and (b) the fact that the diameter – quantified using the average shortest distance between two nodes – increases logarithmically with the number of the nodes of the network (Amaral et al., 2000; Watts & Strogatz, 1998). Several network cases have been found to be eligible to be characterized as small world networks such as social networks, citation networks and chemical reaction networks.
Based on the abovementioned, in order to characterize a network as a small world networks, the network should illustrate a small degree of separation property (average shortest path size between all nodes) and a clustering property (nodes are clustered with in-between connections forming triangles) – estimated using Equation 1.
1 (1)
As pointed by Albert & Barabási, (2002), random networks also illustrate the property of small degree of separation, yet they do not illustrate the clustering property. As such, the measure that diversifies a small world network from a random network is:
≫ (2)
where is the average clustering coefficient (Equation 3) of the examined network and is the average clustering coefficient of a random network.
1
(3) The average clustering coefficient of a random network is estimated given the average network degree ( and the total number of nodes ( ):
/ (3)
The average path length for a random network is estimated:
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Scale-free networks are investigated and characterised based on the distribution of the degree of small world networks (Barabási & Bonabeau, 2003). This type of networks emerged from a study on the World Wide Web (Barabási & Bonabeau, 2003) and has found to apply as a sub-category of small world networks together with broad-scale networks, and single scale networks (Amaral et al., 2000). The real world network cases found to fall into the scale-free networks category (such as World Wide Web, citation networks, airport networks, metabolic reaction networks) exhibit a degree distribution that decays as power law distributions ( ~ where is the probability the degree of the network to be equal to the random degree ). Broad-scale networks were found to follow a power-law distribution which is followed by “a sharp cutoff” and single scale networks characterised by a degree distribution of a fast decaying tail (Amaral et al., 2000).
Closely to the characterization of networks, scientific research also focus on the use of the graph measures that have been developed to analyse networks and the development of new measures that might be indicative of the network characteristics and their performance. The most widely used fall into the centrality (Freeman, 1978) and vulnerability measures categories.
Graph ( , ): A set of nodes ( ) connected with a set of links ( ) used to represent a network.
Adjacent: Two nodes that are directly connected.
Average Shortest Length: The average shortest length of a network is defined as
1 1
, (1)
Node Degree (or point centrality) : The number of neighbouring nodes to which a node is connected.
Closeness Centrality: Let , be the length of shortest path between two nodes. Closeness centrality of the random node is defined as:
, (4)
Closeness centrality illustrates the number of
Betweenness centrality: The summation of the ratios of the number of shortest paths that pass through the node examined ( ) for the random pair of nodes , to the total number of shortest paths between the two random nodes for all random nodes of the network.
,
(3) In most cases, networks are being investigated to be identified in a network class such as small world network. The graph measures have been used in some transportation related measures with extensive work to take place on the analysis of sea transportation networks
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(Ducruet et al., 2010; Hu & Zhu, 2009), air transport networks (Amaral et al., 2000; R Guimerá & Amaral, 2004; Roger et al., 2003; Li & Cai, 2004) and railway transportation networks (Gonçalves et al., 2009; Ouyang et al., 2014). On spatial transportation networks such as road networks complex network analysis and the use of graph measures is limited mainly due to the spatial constrains that road networks impose (Barabási & Bonabeau, 2003; Erath et al, 2009; Xie & Levinson, 2007). Barabási & Bonabeau (2003) used the example of the US highway to illustrate the randomness of the road network where the degree distribution follow a binomial distribution. Xie & Levinson (2007) argued that the analysis of complex networks is node-centric while road networks are link-centric and suggested three measures for structural analysis of networks (heterogeneity, connection pattern and continuity). Erath et al. (2009) investigated the development of the Swiss road and railway network in 50 years on centralities, saturation, efficiency and density and found regular network patterns for the network. On the other hand, (Jiang, 2007) found that urban road networks illustrate a small world structure with scale-free properties by examining 40 cities in the USA, while (Sun, 2012) presents a tutorial on how to examine the existence of the structure and the properties showing that the examined city (Avignon) has a small world structure with scale-free properties. Public transport has also received attention on the categorization of structure and definition of properties. Von Ferber, Holovatch, Holovatch, & Palchykov (2008) examined public transport networks from 14 large cities using graph properties analysing it in 3 different spaces based on the way links are modelled (stop to stop connections, directly connected stops connections, routes).
In this paper we present the findings on the complex structure and properties of the road transportation network and the public transport network of the metropolitan area of Thessaloniki and the road transport network of Europe. We start by identifying the structure of the two networks and then we apply graph measures.
2. Networks Structures and Properties
In order to investigate the controversial findings on the random or small worlds structure of road networks we examine 3 networks that were available and the number of links and nodes could be handlled by the analysis tool (Matlab). The choisen networks used are presented in Figure 1. This choice was made in order to examine different network hierarchies in the Greek also Public Transport (PuT) in order to examine the effect it has on the networks structure and properties.
First we present the European Highway network (Figure 1a) that was extracted from the TransTools model (Rich et al., 2009). The network consists of 23347 nodes and 35248 links. The findings concerning the network characterization are presented in Table 1. As it is clearly evidenced the hypothesis of Europe’s highway network being a small world network should be rejected as while ≪ the average length of the network ( ) is much higher than the average path length of the random network ( ). Furthermore, the hypothesis that the Europe’s highway network could illustrate scalefree properties was rejected as the -value of the Kolmogorov-Smirnov test for fitting a power law distribution on the degree distribution was found to be equal to zero and the log-log plot of the degree cumulative distribution illustrates an exponential decay.
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(a) Europe Highways (b) Thessaloniki Road (c) Thessaloniki PuT
Figure 1: Networks for network structure and properties examination
The next network examined is the road transportation network of the Thessaloniki city in Greece consists of 312 nodes and 1015 links covering the metropolitan area of the city (Figure 1b). As it is clearly evidenced the clustering coefficient of the network is much higher than the clustering coefficient of a random network with the same characteristics suggesting that although the average path length is higher than the average path length of a random network, the network could be considered a small world network. On the examination of the hypothesis concerning scale-free characteristics, it was found that the network of Thessaloniki, cannot be considered a scale-free network, as the -value of the Kolmogorov Smirnov test for power law distributions is equal to zero.
Finally, the Public Transport Network of Thessaloniki used was modelled in the L Space defined by (Von Ferber et al., 2005) where each stop (node) is directly connected to on stop only if there a direct connection that connects the latter to the first. As such, an intermediate stop in the network is directly connected to the starting stop of the route. The network consists of 130 lines and 2250 nodes (stops) that in L space were modelled as 3979 links. As it is clearly presented in Table 1, the PuT network of Thessaloniki is characterized by a rather high clustering degree that is much higher than the degree of random network with the same average degree. Furthermore, the average length –although higher—is within acceptable margins for characterizing as a small world network. Another interesting feature of the PuT network of Thessaloniki is that although there is a clear exponential decay the data could be fitted partly by a power-law distribution with 3.140. This type of networks have been identified by Amaral et al. (2000) as broad scale network as “the distribution follows a power law regime that is followed by a sharp decay”.
Table 1: Network Characteristics for Small World Network Structure Identification
Network
Europe Road 3.019 82.743 9.021 0.0818 0,0001
Thessaloniki Road 2.971 20.091 6.375 0.2466 0.0029 Thessaloniki PuT (L space) 106.000 7.928 1.656 0.479 0.0471
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3. Graph Network Measures
Apart from the examination of the structure and properties we investigate the usability of the graph measures in the Greek urban context of Thessaloniki for both the road network and the PuT network. The findings for the two examined networks are presented in Figure 2 and Figure 3 for Thessaloniki road network and Thessaloniki PuT network respectively.
The figure bellow (Figure 2) illustrates the performance of the transportation network given only the node-centric analysis and without taking into account demand or any other transportation related metric. More specifically, Figure 2a presents the degree of each node which is a measure of connectivity. Betweenness centrality (presented in Figure 2b) shows the nodes that are mostly “in-between” other pairs of nodes when accounting for the shortest route. This measure could be used to identify the nodes that would receive the highest demand and as a consequence the nodes which could be characterised as hubs. As it is illustrated in Figure 2c the nodes with the highest Closeness Centrality are those being in the densest areas of the network. This measure allows for a quick identification of the nodes which are central and can be illustrate the nodes which are most probable to be in the centre of the network given different cost functions which could define closeness.
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In a similar way, the PuT network of Thessaloniki has some interesting results to present. With the use of Degree and betweenness centrality it is rather easy to identify the hubs when modelling PuT in the L-space. Furthermore, it is presented that roads with high PuT connectivity are indicates with a high betweenness. On the contrary, closeness centrality does not indicate the most central nodes of the PuT network, but only those that are closer to all other nodes. This could happen in cases the distance between two stops is rather small or there are many stops that are connected directly together.
Figure 3: Graph Measures for the Thessaloniki PuT network
4. Discussion Future Work and Conclusions
This paper presented the application of complex network analysis and graph theory measures for transportation networks. In the beginning the theory of complex networks and the graph theory measures are presented and explained. In the complex network analysis the networks did not share the same structure and properties and in a way confirmed the fact that there is not a clear structure that could govern the way transport networks are build and there is not a clear universal property that those networks share. On the other hand it was found that the Thessaloniki road and PuT network could be characterized as small world, a structure that is being explored for its special characteristics. This could allow for advances in the premises of
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this research that would benefit transportation modelling on better identification of systems vulnerability, propagation of an incident’s impact and others.
In graph measures analysis the Road and the PuT Network of Thessaloniki was used to estimate degree, closeness centrality and betweenness centrality. Although their indications are clear by the application here, it is believed that the combination of those measures with existing metric such as accessibility could allow the definition of better performance indicators for the transportation system. Especially on the evaluation of PuT network, it could be important to use those measures in conjunction to other characteristics such as demand and congested travel time.
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