Abstract
Purpose
The purpose of this paper is to explore the evolving mechanism of urban roadway network. With the consideration of self-organization effect and planning effect during evolution, the authors try to demonstrate the impact of preferential attachment, module scale and module structure on the evolving network model.
Design/methodology/approach
The roadway network is built in the form of abstract network by dual approach. By using the evolving model of modular growth, the authors analyze the effects and mechanism of the evolving process. Then through numerical analysis, the impact of evolving effects on urban roadway network topology structure is discussed from the aspects of preferential attachment, module scale and module structure.
Findings
The module structure property, small-world property and scale-free property of roadway network can be affected with various degrees by the change of preferential attachment and module scale. However, the impact of module structure on network properties is small, which can be ignored. Therefore, in practice, the self-organization effect and planning effect of evolving network can be reached by changing the preferential attachment and module scale, so as to generate the network structure with specific properties.
Research limitations/implications
Some local events, such as road extensions, road demolition and intersection rebuilding, exist during the evolving process under real-world situation. While those cases have not been considered in preferential attachment. Therefore, researchers are encouraged to take these factors into consideration in further research.
Practical implications
The paper has implications for practice in urban transportation planning and roadway constructions, which can help to guide the planning of urban roadway and to adjust or restore partial network when broken down according to the evolving law.
Originality/value
The impact of preferential attachment, module scale and module structure on the evolving network model is measured. And the relationship between different network properties can be used to build some patterns of network. From this point of view, the development of urban roadway network can be predicted and intervened.
Keywords
Citation
Zou, Z., Liu, P., Zhou, S., Xiao, Y., Xu, X. and Gao, J. (2015), "Analysis on evolving model with modular growth of urban roadway network topology structure", Kybernetes, Vol. 44 No. 4, pp. 505-517. https://doi.org/10.1108/K-12-2013-0272
Publisher
:Emerald Group Publishing Limited
Copyright © 2015, Emerald Group Publishing Limited
1. Introduction
Urban roadway network is the basis of a city’s development and the main carrier of urban transportation activities. Recently, many researchers have been devoted to the studies of network complexity in urban roadways, especially on the structural complexity, the spatial-temporal distribution complexity, the evolving mechanism of road network, which are the main issues in those studies.
Transportation development is a complex and dynamic process that involves a magnitude of dimensions, which may be topological, morphological, technical, economic, managerial and political. Despite the fact that the growth of transportation networks has been decades, it may still be tractable and predicable with a further understanding of the underlying mechanism. With this thought, sustained efforts have been made in the modeling and analysis of transportation networks (Levinson and Yerra, 2006). A capacity-related reliability for transportation networks with random link capacity was introduced by Chen et al. (1999). Due to large variability associated with link capacities, a probabilistic approach was adopted in their study to model the different physical and operational factors that often degrade the capacity of roadways. An empirical analysis of transit network evolution was put forward, and the results showed that supply increases with demand and population density, and decreases with the number of schoolchildren in the vicinity (Mohammed et al., 2006). The topological evolution of surface transportation networks was presented and discussed by Xie and Levinson (2007), by using empirical evidence and a simulation model. Besides those, some special issues on the evolution of transportation network infrastructure also have been discussed by Levinson (2009). A state-of-the-art review of the transport network design problem (NDP) under uncertainty was presented, and some new developments on a bi-objective-reliable NDP model were also studied (Chen et al., 2011), which explicitly optimizes the capacity reliability and travel time reliability under demand uncertainty. The prioritization of the association of South East Asian Nations highway network development was evaluated by Lee et al. (2011). Also, the implications of existing trends on future network investment were analyzed (Levinson et al., 2012).
The studies above analyzed the characteristics of transportation networks from the macroscopic perspective, the inner properties from microscopic perspective are important as well. Although the topological property of urban roadway network has some similarities with that of other complex networks, it still has its own unique features, such as selectivity and autonomy, etc. Combined with the features, the evolution of urban roadway network topology structure is investigated, which forms a good foundation for further study on the structural layout of the network, roadway network carrying capacity, cascading failure process and urban traffic congestion alleviation (Zhao et al., 2006).
To study the complexity of urban roadway network, the topology structure is built in the form of abstract network. At present, there are two main ways of building the abstract network, primal method and dual approach (Jiang and Claramunt, 2004). The former takes the intersections and all roads as the set of nodes and links, respectively, while the latter considers the roads as vertices and intersections as links. In this paper, the urban roadway network topology structure is built by the dual approach as study object to analyze the effects and mechanism of the evolving process by referring to the evolving model with modular growth (Zou et al., 2012). Meanwhile, from the aspects of preferential attachment, module scale and module structure, the impact of evolving effects on the topology structure is also discussed.
2. The evolving effects and mechanism of urban roadway network
2.1. The evolving effects of urban roadway network
From the microscopic perspective, the evolution of urban roadway network is driven by multiple factors (Kauffman, 1993), which can be regarded as the result of self-organization effects. From the macroscopic perspective, the evolution is carried out stage by stage, and the active participation of the decision makers forms the planning effects. In this paper, the evolving effects of urban roadway network are divided into self-organization effect and planning effect.
In the process of city development, urban roadway planning is frequently adjusted. The original roads are either demolished or rebuilt, while those newly built roads become parts of the network as planned. On one hand, the evolution of roadway network needs to match with the self-organization effect caused by the variation of travel demand, land-use adjustment and the social economic development. On the other hand, the man-made factors in planning process also help to shape the network. Therefore, the formation of modern urban roadway network is the consequence of co-evolution under self-organization effect and planning effect.
2.2. The evolving mechanism of urban roadway network
The evolving effect of urban roadway network topology structure exhibits the corresponding evolving mechanism. By adopting the evolving process of abstract network for reference (Barabasi and Albert, 1999), this paper holds that the evolution of urban road network topology structure follows the mechanisms of preferential attachment and modular growth. Preferential attachment mechanism reflects the self-organization effect, indicating that the newly planned roads prefer to intersect with the existing roads which have stronger attraction, while the modular growth mechanism refers to the co-effect between self-organization and planning. Through self-organization, many roads form a module with the corresponding module structure, and then this module is planned as one part of the urban roadway network.
2.2.1. Preferential attachment mechanism
Preferential attachment mechanism reflects the self-organization effect of urban roadway network topology structure. In the process of network growth, the new roads tend to intersect with those with priorities (such as higher traffic flow or more intersection nodes, etc.). After they are added into the network, the attached roads burden more volume. And the number of intersections will also be increased, thus giving the roads more priorities. Finally the network reaches to some certain level, where “the rich get richer.”
2.2.2. Modular growth mechanism
The modular growth mechanism reflects the compound effect of self-organization and planning of roadway network topology structure. Different from the single-node growth model of abstract network, the evolving rate of urban roadway network is much greater. At each time step, several roads will be planned to become parts of the network, and those newly added roads, which have some inner relationship, perform as a module. In the evolving process of network topology structure, we mainly focus on two aspects to study the module structure: one is the module scale, which reflects the planning effect of the structure; the other is the module structure, exhibiting the self-organization effect.
a. Module scale
The module scale refers to the number of nodes and links in the abstract network, which reflects the detailed degree of network division, as well as the planning effect of topology structure’s evolving process. The larger the module scale is, the stronger the planning effect is, and vice versa.
b. Module structure
Module structure is the initial network form before the new module is added in. The new module may display different network morphologies in the topology structure. Since the new module is small, its morphologies are not always significant. The formation of newly added module may sometimes be random, while the randomness has a positive relationship with the degree of self-organization effect. That case is very similar with small-world network. Referring to the study of the module growth model (Zou et al., 2012), we regard the newly added modules as the small-world network in the field of complex network, which lies between the regular network and the random network.
2.3. Evolving process of urban roadway network with modular growth
Referred to the abstract network model with modular growth, a module composed of several nodes is added to the urban roadway network at each time step during the evolving progress. The added module will be transformed into a small-world network, and after that, the nodes of the new module would be connected to the nodes of the initial network by the rule of preferential attachment. The progress is illustrated in Figure 1.
As shown in Figure 1, the number of nodes in the initial network is e 0=6, and number of links is m 0=9. In each module, the number of nodes is s=10, and the average degree of module nodes is K=4. The new module would be connected to some nodes of the initial network, and the number of connections is m=5.
Based on the mechanism of preferential attachment, the nodes of the new module are selected that the nodes with larger degree would be connected to the existing nodes, and the nodes selected to be connected are also determined by the rule of preferential attachment. The specific evolving process of the proposed network model is presented in Table I.
3. Analysis of impact on topology properties of urban roadway network
3.1. Properties indices
Four indicators (modularity measure, degree distribution, clustering coefficient and average path length) in the complex networks theory are adopted to study the change of network properties, from the point of view of preferential attachment, module scale and module structure. The dependences of the modular structure property, the scale-free property and the small-world property of the networks can be analyzed to provide references for measuring the self-organization effect and planning effect in evolving network.
The four indicators are as follows.
Modularity measure Modularity measure is proposed to describe the strength of module structure in a network according to Newman and Girvan’s (2004) study: (Equation 1) where n is the number of inner-module links, and m is the number of connections between newly added module and the existing network.
Degree distribution Since the degree distribution of scale-free network nodes obeys the power-law distribution, we can judge if a network has the scale-free property. An analytical function of the degree distribution is developed in the view of the mean field theory (Zou et al., 2012), from which we can obtain the probability P(k) that the degree of node i in the network is equal to k: (Equation 2) where A=2n/m+2, and (Equation 3), which contains factorial expression. And the whole derivation procedures can be found in the Appendix.
Clustering coefficient This indicator is a measure of the degree to which nodes in a network tend to cluster together. In the network, for a selected node i, there are at most k i links linked to other nodes. Among the k i links only E i links actually exist. The clustering coefficient C of the whole network with N nodes is: (Equation 4)
Average path length The average path length L is defined as the average value of the all paths’ lengths between any two nodes in a network and expressed as: (Equation 5) where N is the number of the network’s nodes. And d ij refers to the path length between node i and node j.
We can judge whether the network has the small-world property by analyzing the variation tendency of clustering coefficient and average path length with the growth of network, since small-world networks generally have the property of large clustering coefficient and small average path length (Albert and Barabasi, 2002).
3.2. The impact from the preferential attachments
Preferential attachment shows the self-organization effect of evolving network, and the effect under different number of attachments can reflect the impact on network properties referred to the strength of self-organization. At each time step, the connection number m of the new module to be added to the existing network is variable, and the degree of self-organization can be changed by adjusting the value of m. The larger the value of m, the stronger the effect of preferential connection and self-organization.
3.3. The impact from the module scale
The module scale indicates the planning effect of evolving network, and different module size can measure the impact on network properties under planning effect. As the model proposed by the paper, the module size can be changed by adjusting the nodes number s and the average nodes degree K of the newly added module.
3.4. The impact from the module structure
The module structure indicates the self-organization effect of evolving network. For the added small-world module, the randomness can be changed by adjusting the rewiring probability α. The impact on network properties of different randomness reflects the influence of self-organization effect. The bigger α is, the stronger this effect is.
4. Numerical analysis
4.1. Initial network
When the time step is large enough, the size of initial network has no obvious impact, and even can be neglected. Assume that in the initial network in Figure 1, the number of nodes is e 0=6, and that of links is m 0=9.
4.2. Index analysis
During the course of selecting the indicators, degree distribution, clustering coefficient and average path length, various parameters need to be chosen according to different situations, which should reflect the following three influencing factors, preferential attachment, module scale and module structure. Table II gives the details:
N is the network size, which is the total number of the network nodes after the evolving process is completed;
m is the connection number between the newly added module and the initial network;
s is the node number of new added module, K is the average degree of the module, the link number of it is n=K×s/2; and
α is the probability of link’s random reconnection in the initial module (see step 1 in Table I).
4.3. Results and discussion
4.3.1. Preferential attachment impact
From the analysis it can be found that modularity measure is reduced with m increased, and the values of network modularity measure Q are equal to 0.8, 2/3, 0.5 and 1/3 for m=5, 10, 20, 40, respectively.
As shown in Figure 2, with the increasing m, the regularity of numerical results for degree distribution is raised, and the distribution curve tends to become stable, which indicates the increasing of scale-free property and small-world property in the proposed model.
Contrarily, as shown in Figure 3, with the increase of m, the values of clustering coefficient and average path length decrease, while the variation trend on the network scale remains the same. The change of small-world property could not be determined since the variation range of the clustering coefficient and average path length is quite the same.
4.3.2. Module scale impact
Modularity measure is raised with s and K increased. The values of network modular measure Q are equal to 3/13, 0.375 and 2/3 for s=6, K=2; s=4, K=4; s=14, K=6, respectively.
As shown in Figure 4, the regularity of numerical results for degree distribution decreases. The distribution curve tends to become tightened, which indicates the reduction of scale-free property.
On the other hand, as shown in Figure 5, the values of clustering coefficient and average path length increase, but the trends on the network scale keeps unchanged. The rangeability of the clustering coefficient is large while that of the average path length is small, so the small-world property will be increased with the rising s, K.
4.3.3. Module structure impact
Modular measure Q≈2/3 when the value of α is changed while m and n remain unchanged.
As shown in Figure 6, the variation of degree distribution curve is negligible with α increasing.
As show in Figure 7, the variation of clustering coefficient and average path length changes a little with α increased, which implies that the small-world property of this model also changes a little.
5. Conclusions
This paper investigates the impact of preferential attachment, module scale and module structure on the evolving network model. The results show that the inherent properties (module structure property, small-world property and scale-free property) remain unchanged, but the strength of these inherent properties can be varied by adjusting the values of modularity measure, clustering coefficient and average path length.
Shown from the analysis above, the increasing of preferential attachment property may decrease the network module structure property, and increase the small-world property and scale-free property. The expanding of module scale may reduce the network scale-free property, and increase module structure property and small-world property. Additionally, the variation of module structure has little impact on the three properties.
These changes of the properties mentioned above can clearly reflect the role of self-organization effect and planning effect playing in the evolving process of urban roadway network topology structure. The self-organization effect of evolving network can be embodied in preferential attachment and module structure, while the planning effect can be reflected in module scale. However, the impact of module structure on network properties is small, which can be ignored. Therefore, in practice, the self-organization effect and planning effect of evolving network can be reached by changing the preferential attachment and module scale, so as to generate the network structure with specific properties.
Appendix
The derivation procedures of probability function P(k) described in Section 3.1 are as follows.
First, we assume that the change of node degree is continuous. If the time step is large enough, the size of initial network can be negligible. By using the mean-field theory, the change rate of node degree can be expressed as:
Let k i (t i )=k x , where k x represents the initial degree of node i belonged to the new module after being added to the network. Then:
The probability density function at time t i is P i (t i )=1/(t+e 0/s), so the probability that degree of node i is k can be:
where k x k.
The probability that the initial degree of node i is k x depends on the small-world evolution and preferential mechanism in the new module. According to the Newman and Watts’s (1999) study, when α=0, all links of the new module would not be rewired, thus degree distribution is the same as regular network and all nodes’ degree is K. When α > 0, since one node of each link is kept unchanged, each node is linked by at least K/2 links after rewired. Therefore, after small-world evolution, the probability that the degree of node i in the new module is k′ can be expressed as:
Then the nodes in the newly added module will be chosen preferentially for m times. The probability that node with degree k′ can be chosen for k′′ times (namely node degree is increased by k′′, k′′m) is:
Thus, the probability that the initial degree of node i from new module is k x can be obtained as:
From the analysis above, the probability of node i degree being k is:
where A=2n/m+2 and (Equation 13). For the derivation result of f(k) is too complicated to represent, so we keep the form of ∂f(k)/∂k in the formula P(k).
Corresponding author
Jianzhi Gao can be contacted at: [email protected]
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Further reading
Gao, Z.Y. and Li, K.P. (2005), “Evolution of traffic flow with scale-free topology”, Chin. Phys. Lett , Vol. 22 No. 10, pp. 2711-2714.
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Strogatz, S.H. (2001), “Exploring complex networks”, Nature , Vol. 410 No. 6825, pp. 268-276.
Wu, J.J. , Gao, Z.Y. , Sun, H.J. and Zhao, H. (2010), Urban Transportation System Complexity--Complex Network Method and Its Application , Vol. 5, Science Press, Beijing, p. 391 (in Chinese).
Zanin, M. , Lacasa, L. and Cea, M. (2009), “Dynamics in scheduled network”, Chaos: An Interdisciplinary Journal of Nonlinear Science , Vol. 19 No. 2, p. 259.
Zhao, Y. , Du, W. and Chen, S. (2009), “Application of complex network theory to urban transportation network analysis”, Urban Transport of China , Vol. 7 No. 1, pp. 57-65.
Zou, Z.Y. , Xiao, Y. and Gao, J.Z. (2013), “Robustness analysis of urban transit network based on complex networks theory”, Kybernetes , Vol. 42 No. 3, pp. 383-399.
Acknowledgements
This study is supported by the National Natural Science Foundation of China (No. 51078165). The authors would like to thank anonymous referees for their helpful comments and valuable suggestions which improve the content and composition substantially.