- Email: [email protected]
Dynamics of traffic networks: From microscopic and macroscopic perspectives
Physica A 387 (2008) 1648–1654 www.elsevier.com/locate/physa
Dynamics of traffic networks: From microscopic and macroscopic perspectives H.J. Sun ∗ , J.J. Wu, Z.Y. Gao State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, PR China School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, PR China Received 25 April 2007; received in revised form 7 July 2007 Available online 20 November 2007
Abstract Considering the microscopic characteristics (vehicle speed, road length etc.) of links and macroscopic behaviors of traffic systems, we derive the critical flow generation rate in scale-free networks. And the dynamics of traffic congestion is studied numerically in this paper. It is shown that the queue length increases with microscopic characteristics of links. Additionally, the critical flow generation rate decreases with increase of the network size N , maximum speed vmax and parameter τ . The significance of this finding is that, in order to improve the traffic environment, both the local information for the single link and behaviors of the whole network must be analyzed simultaneously in a traffic system design. c 2007 Elsevier B.V. All rights reserved.
PACS: 89.75.Hc; 89.75.Fb; 89.20.-a Keywords: Dynamics; Traffic networks; Complex networks; Microscopic and macroscopic
1. Introduction The structure and dynamics of complex networks have attracted a tremendous amount of recent interest [1–3] since the seminal works on scale-free networks by Barab´asi and Albert [4] and on the small-world phenomenon by Watts and Strogatz [5]. Mathematically, a way to characterize a complex network is to examine the degree distribution P(k), where k is used to measure the number of links at a node. Scale-free networks are characterized by P(k) ∼ k −λ , where k is the algebraic scaling exponent [4]. Recently, more and more researchers have begun to develop models for explaining the dynamic behaviors of traffic on complex networks, i.e. the critical value of the flow generation rate and the cascade failure [6–8]. A detailed analysis of dynamic behaviors caused by traffic congestion in gradient networks suggests that there exists a critical value of the average degree. For values of the average degree below this critical value, large scale-free networks are somewhat more prone to congestion than random networks with the same number of nodes and average degree while the opposite is true above the critical value [9]. In particular, Zhao et al. [6] propose a cascade failure model for complex networks and
∗ Corresponding author at: State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, PR China.
E-mail address: [email protected] (H.J. Sun). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.10.054
H.J. Sun et al. / Physica A 387 (2008) 1648–1654
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uncovers a congestion phase-transition phenomenon in terms of the key parameter characterizing the node capacity. Wu et al. [10] derive the dynamic critical flow generation rate based on the unoccupied capacity in scale-free networks. A common feature in previous studies is that they don’t consider the microcosmic characteristics of traffic problem such as road length, vehicle speed, density etc. Of course, there are many studies on microcosmic traffic flow models [11–13], in which traffic phenomena on a single link are analyzed, while traffic behaviors on the whole network are not considered. The quantity of interest is the critical rate Rc of flow generation (as measured by the number of flows created within the network in unit time), at which a phase transition occurs from free to congested traffic flow. For R < Rc , the numbers of created and delivered flows are balanced, resulting in a steady state, or free flow of traffic. The network is in an uncongested state, while for R ≥ Rc , congestions occur in the sense that the number of accumulated flows increases with time, due to the fact that the capacities of links for delivering flows are limited. We are interested in determining the phase-transition point Rc , given a network topology. Inspired by previous work [11], we study both microscopic behaviors of individual vehicles on links and the statistical properties of traffic networks. In this paper, the main point is deriving the critical flow generation rate of the traffic network based on the microscopic characteristics. Further we study how the statistical properties, both at microscopic (road length, vehicle speed, etc.) and macroscopic levels (behaviors on the whole network), vary with the flow generation rate R. 2. Model of critical flow generation rate In this section, we derive the critical flow generation rate by recalling the model proposed by Mahnke and Pieret [11] first. For complete details, we refer the reader to the reference. For succinctness, we present the notation used for this model as follows. S: the mean size of the traffic congestion or the queue length of traffic congestion; F: the total number of vehicles; L: the total length of the road; l0 : the effective length of a single vehicle; τ : the average time of the head vehicle changing its state from congestion to free; d: a positive control parameter, which can be seen as a characteristic headway for the transition between noninteracting and interacting phases; vmax : the maximal speed allowed. In Ref. [11], a stochastic description of congestion formation using the master-equation approach is given, and the transition probabilities for the jump processes are constructed. The following equation holds: S=F−
L/l0 − F p . (vmax τ/2l0 ± (vmax τ/2l0 )2 − (d/l0 )2 )
(1)
The queue is formed when S ≥ 0 and the traffic congestion occurs. Therefore, the critical flow generation rate Rc can be obtained when S = 0. To account for the network topology, we assume that the capacities for processing flows P are different for different links, depending on the degree of nodes connecting this link. Then, Pi j = β(ki + k j ), where Pi j is the processing capacity of link i − j, ki is the degree of node i and β is the parameter. In given manual networks, starting from an unloaded network, the evolving mechanism of traffic flow can be described as: (1) Generation traffic flow. Assumed that the flows are generated at the nodes. At each time step, we impose a constant input of newly created traffic flow R. The source of each flow as well as its destination is chosen at random among all the nodes of the network. Besides, each node can send P flow which is related to the degree of the node at each time step and, as a consequence, one node, i, can have a queue to be delivered. (2) Movement. Flows move through the graph simultaneously searching for their respective destinations based on the shortest path algorithm. At each time step, only at most Pi j flows can be transported on link i − j according to the FIFO (First-In–First-Out) principle. When the queue at a selected link is full, the link won’t accept any more flows and the flow will wait for the next opportunity. Once a vehicle arrives at its destination, it will be removed from the system [14]. Because the link with the largest betweenness can be easily congested and the congestion can quickly spread to the entire network, it is necessary to consider only the traffic balance of this link. Since the flows are transmitted along the shortest paths from the sourcePto the destination, the probability that a created flow will pass through the link with the largest betweenness i is Bi / j B j [6]. At each time step, on average, R flows are generated. Thus, the average
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number of flows that the link with the largest betweenness receives at each time step is X Bj, F = R · N · hDi · Bmax /
(2)
j
where hDi is the average shortest path length of the network, N is the number of nodes in the network, and max is the index of the node with the largest betweenness [6]. We use the Eqs. (1) and (2) to derive the critical flow generation rate Rc . Let q A = vmax τ/2l0 ± (vmax τ/2l0 )2 − (d/l0 )2 , and C = L/l0 . Thus, F=
C−F , A
(3)
C . A+1
(4)
and, then, F=
Based on Eq. (1), the following equation holds: C Bmax = R · N · hDi · P . A+1 Bj
(5)
j
P
For the network, since j B j = R · N · hDi, thus we can calculate the critical flow generation rate Rc : P C· Bj Rc =
j
(A + 1) · N · hDi · Bmax
.
(6)
Up to now, we have obtained the critical value of the flow generation rate Rc , which can be used to describe the flow dynamics on networks. For a small value of the generation rate R, the flow assigned to the link is small so it can be processed in unit time. But for a larger R, the flow may exceed the link’s processing capacity, and it is more likely to accumulate at this link, resulting in traffic congestion. If the congestion cannot be alleviated, it breaks the equilibrium of the network and quickly spreads to the entire network. Therefore, it is important to determine the value of Rc so as to avoid and control the congestion effectively. 3. Simulation results Recently, there has been much research on the representation of urban networks [15–20]. After an in-depth investigation of both the topological (dual) graph [16], where streets are nodes and intersections are edges, and the spatial (primal) [17] graph, where intersections are turned into nodes and streets into edges, representations of street networks, Scellato et al. [18] have provided a tool for the analysis of the backbone of a complex urban system represented as a spatial (planar) graph. Kurant and Thiran [19] extracted the real physical topology from timetables. Jiang and Claramunt [20] proposed a topological analysis of large urban street networks based on a computational and functional graph representation. This representation gives a functional view in which vertices represent named streets and edges represent street intersections. While in this paper, links and intersections corresponds to the edges and nodes in the constructed urban traffic network. Generally, we represent networks as graphs G = (V, E), where V and E are the set of vertices and edges respectively. G is described by the adjacency matrix ei j . Define N as the size of the network. If there is an arc between node i and j, the entry ei j is the value 1; otherwise ei j = 0. In this paper, we use the Molloy–Reed algorithm [21] to generate scale-free networks. Selection of these particular structures is based on two reasons: (1) Scale-free is a most popular structure in the real world. A larger number of previous literature studies have concluded that most real networks display scale-free characteristics, including the Internet, WWW networks, power grids and so on [5,22,3];
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Fig. 1. The queue length S as a function of time steps for different flow generation rates R. The parameters are N = 600, L = 600, l0 = 8, d = 20, τ = 3, vmax = 20, β = 0.05.
(2) In our previous work [14], we find that the urban transit networks exhibit a power-law distribution with exponent λ ≈ 2.24. The congestion in scale-free networks is studied using computer simulation and the results are presented in this section. All results presented in this paper are for undirected, uncorrelated scale-free networks with an exponent of the power-law degree distribution λ ≈ 3 generated using the generation method mentioned above with hki ≈ 3. Using this exponent is based on the following reason: In classical references, it is obtained that most real-world networks follow power laws with exponent range 2–3. Meanwhile, the seminal work by Barab´asi and Albert (BA model) has constructed a scale-free network with exponent 3. Therefore, carrying out our study with exponent 3 makes it more universal and easier to generalize the results. It is assumed that at each time step, 0.1 unit flow is created and there are 0.1 × N flows generated in the whole network, and the parameter β = 0.05 in our paper. The simulation process is based on the evolving mechanism in Section 2. Throughout the paper, for each network, more than 25 realizations are averaged. Fig. 1 shows the time evolution of the queue length of traffic congestion S for different critical flow generations rate R on the link with the maximum betweenness. We see that, for all cases considered, S is zero at the beginning, and it suddenly increases when time is larger than a critical value. We also observe that S will increase with R, which means that the addition of flow can cause possible congestion that is most likely to occur at heavier links. As a result, phase transition can be delayed in the sense that the network can be more tolerant to traffic congestion for smaller values of R. Clearly, in comparison with the small R, the heavy flow on a central road (the edge with the highest betweenness) is difficult to process for larger R. Therefore, more and more traffic flow is accumulated and the traffic congestion occurs. In Fig. 2, we report the simulation results for S as a function of the maximum speed allowed, vmax . As one can c see, first, as vmax increases, S increases gradually; only when vmax is increased above a value vmax (in this paper, c c vmax ≈ 14) is a suddenly jump observed. This result suggests that, to a certain extent, decreasing vmax may relieve the traffic congestion. However, the larger of the maximum allowed speed vmax , the larger the headway is and the longer the time for the head vehicle to leave the queue. Thus, the queue length on links will increase. Fig. 3 shows the relationship between S and the total length of road L for the flow generation rate R = 0.1. The monotonic increase of S can be well approximated by the linear relation S ≈ 0.023L + 0.0028. This means that for a longer road, more drivers alternate it as the better trip route. As a result, this road is easy to be distributed flows and will become congested. Next, we compute the critical flow generation rate Rc and plot the relationships between Rc and τ , Rc and vmax in Figs. 4 and 5 respectively. The common characteristics of these two figures are that the critical generation rate Rc is decreasing with the increase of network size N . These figures also show that parameters τ and vmax have the same effects on the falling shapes of Rc for different network sizes N . However, the larger the network size is, the quicker
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Fig. 2. The queue length S as a function of the maximum allowed speed vmax with flow generation rate R = 0.1. Other parameters are N = 600, L = 600, l0 = 8, d = 20, τ = 3, β = 0.05.
Fig. 3. The queue length S as a function of the total road length L with flow generation rate R = 0.1. Other parameters are N = 600, l0 = 8, d = 20, τ = 3, vmax = 20, β = 0.05.
of Rc decreases, which means that for a larger traffic network, by only controlling microcosmic characteristics such as the speed, length of a single road, the passing capacity of the traffic network could not be improved well. In addition, Fig. 4 also shows that, clearly, for a smaller τ , the network size has great effects on Rc . But for a larger τ , Rc will approach a constant value and become independent of τ . This means that, the longer the time that the head car needed from congestion to the free state, the more serious the traffic congestion will be. In Fig. 5, we show Rc as a function of the maximum speed vmax . Our results suggest that the critical generation rate decreases with increase of vmax , which is consistent with the result drawn from Fig. 2. 4. Conclusion In conclusion, in this paper, by taking the microscopic characteristics on a single link into account, we derive the critical flow generation rate in terms of the whole network behaviors. Simulation results indicate that some values of microscopic characteristics on links have large effects on the queue length. And the critical flow generation rate decreases with increase of the network size N , maximum speed vmax and parameter τ . We find that considering only
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Fig. 4. The relationship between τ and critical flow generation rate Rc for different network sizes N = 200, 400, 600. Other parameters are L = 600, l0 = 8, d = 20, vmax = 20, β = 0.05.
Fig. 5. The relationship between the maximum allowed speed vmax and critical flow generation rate Rc for different network sizes N = 200, 400, 600. Other parameters are L = 600, l0 = 8, d = 20, τ = 3, β = 0.05.
the local information for a single link in a transportation design could not alleviate traffic congestion effectively. Thus, behaviors of the whole network must be analyzed simultaneously in order to improve the transport environment. Acknowledgements This paper was partly supported by the National Basic Research Program of China (2006CB705500), NSFC of China (70631001), Foundation for the Author of National Excellent Doctoral Dissertation of China (200763) and Changjiang Scholars and Innovative Research Team in University (IRT0605). References [1] [2] [3] [4] [5]
R. Albert, A.-L. Barab´asi, Rev. Modern Phys. 74 (2002) 47. S.H. Strogatz, Nature (London) 410 (2001) 268. M.E.J. Newman, SIAM Rev. 45 (2003) 167. A.-L. Barab´asi, R. Albert, Science 286 (1999) 509. D.J. Watts, S.H. Strogatz, Nature (London) 393 (1998) 440.
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L. Zhao, Y.-C. Lai, K. Park, N. Ye, Phys. Rev. E 71 (2005) 065105(R). A.E. Motter, Y.-C. Lai, Phys. Rev. E 66 (2002) 065102(R). P. Holme, B.J. Kim, Phys. Rev. E 65 (2002) 066109. K. Park, Y.-C. Lai, L. Zhao, N. Ye, Phys. Rev. E 71 (2005) 065105(R). J.J. Wu, Z.Y. Gao, H.J. Sun, Europhys. Lett. 76 (2006) 787. R. Mahnke, N. Pieret, Phys. Rev. E 56 (1997) 2666. T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331. D. Helbing, Rev. Modern Phys. 73 (2001) 1067. J.J. Wu, Z.Y. Gao, H.J. Sun, H.J. Huang, Mod. Phys. Lett. B 18 (2004) 1043. M. Rosvall, A. Trusina, P. Minnhagen, K. Sneppen, Phys. Rev. Lett. 94 (2005) 028701. S. Porta, P. Crucitti, V. Latora, Physica A 369 (2006) 853. P. Crucitti, V. Latora, S. Porta, Phys. Rev. E 73 (2006) 036125. S. Scellato, A. Cardillo, V. Latora, S. Porta, arXiv:physics/0511063, 2005. M. Kurant, P. Thiran, Phys. Rev. E 74 (2006) 036114. B. Jiang, C. Claramunt, Environ. Plan. B 31 (2004) 151. M. Molloy, B. Reed, Random Struct.Algor. 6 (1995) 161; Comb. Probab. Comput. 7 (1998) 295. [22] A.-L. Barab´asi, R. Albert, H. Jeong, Physica A 272 (1999) 173.
Dynamics of traffic networks: From microscopic and macroscopic perspectives H.J. Sun ∗ , J.J. Wu, Z.Y. Gao State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, PR China School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, PR China Received 25 April 2007; received in revised form 7 July 2007 Available online 20 November 2007
Abstract Considering the microscopic characteristics (vehicle speed, road length etc.) of links and macroscopic behaviors of traffic systems, we derive the critical flow generation rate in scale-free networks. And the dynamics of traffic congestion is studied numerically in this paper. It is shown that the queue length increases with microscopic characteristics of links. Additionally, the critical flow generation rate decreases with increase of the network size N , maximum speed vmax and parameter τ . The significance of this finding is that, in order to improve the traffic environment, both the local information for the single link and behaviors of the whole network must be analyzed simultaneously in a traffic system design. c 2007 Elsevier B.V. All rights reserved.
PACS: 89.75.Hc; 89.75.Fb; 89.20.-a Keywords: Dynamics; Traffic networks; Complex networks; Microscopic and macroscopic
1. Introduction The structure and dynamics of complex networks have attracted a tremendous amount of recent interest [1–3] since the seminal works on scale-free networks by Barab´asi and Albert [4] and on the small-world phenomenon by Watts and Strogatz [5]. Mathematically, a way to characterize a complex network is to examine the degree distribution P(k), where k is used to measure the number of links at a node. Scale-free networks are characterized by P(k) ∼ k −λ , where k is the algebraic scaling exponent [4]. Recently, more and more researchers have begun to develop models for explaining the dynamic behaviors of traffic on complex networks, i.e. the critical value of the flow generation rate and the cascade failure [6–8]. A detailed analysis of dynamic behaviors caused by traffic congestion in gradient networks suggests that there exists a critical value of the average degree. For values of the average degree below this critical value, large scale-free networks are somewhat more prone to congestion than random networks with the same number of nodes and average degree while the opposite is true above the critical value [9]. In particular, Zhao et al. [6] propose a cascade failure model for complex networks and
∗ Corresponding author at: State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, PR China.
E-mail address: [email protected] (H.J. Sun). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.10.054
H.J. Sun et al. / Physica A 387 (2008) 1648–1654
1649
uncovers a congestion phase-transition phenomenon in terms of the key parameter characterizing the node capacity. Wu et al. [10] derive the dynamic critical flow generation rate based on the unoccupied capacity in scale-free networks. A common feature in previous studies is that they don’t consider the microcosmic characteristics of traffic problem such as road length, vehicle speed, density etc. Of course, there are many studies on microcosmic traffic flow models [11–13], in which traffic phenomena on a single link are analyzed, while traffic behaviors on the whole network are not considered. The quantity of interest is the critical rate Rc of flow generation (as measured by the number of flows created within the network in unit time), at which a phase transition occurs from free to congested traffic flow. For R < Rc , the numbers of created and delivered flows are balanced, resulting in a steady state, or free flow of traffic. The network is in an uncongested state, while for R ≥ Rc , congestions occur in the sense that the number of accumulated flows increases with time, due to the fact that the capacities of links for delivering flows are limited. We are interested in determining the phase-transition point Rc , given a network topology. Inspired by previous work [11], we study both microscopic behaviors of individual vehicles on links and the statistical properties of traffic networks. In this paper, the main point is deriving the critical flow generation rate of the traffic network based on the microscopic characteristics. Further we study how the statistical properties, both at microscopic (road length, vehicle speed, etc.) and macroscopic levels (behaviors on the whole network), vary with the flow generation rate R. 2. Model of critical flow generation rate In this section, we derive the critical flow generation rate by recalling the model proposed by Mahnke and Pieret [11] first. For complete details, we refer the reader to the reference. For succinctness, we present the notation used for this model as follows. S: the mean size of the traffic congestion or the queue length of traffic congestion; F: the total number of vehicles; L: the total length of the road; l0 : the effective length of a single vehicle; τ : the average time of the head vehicle changing its state from congestion to free; d: a positive control parameter, which can be seen as a characteristic headway for the transition between noninteracting and interacting phases; vmax : the maximal speed allowed. In Ref. [11], a stochastic description of congestion formation using the master-equation approach is given, and the transition probabilities for the jump processes are constructed. The following equation holds: S=F−
L/l0 − F p . (vmax τ/2l0 ± (vmax τ/2l0 )2 − (d/l0 )2 )
(1)
The queue is formed when S ≥ 0 and the traffic congestion occurs. Therefore, the critical flow generation rate Rc can be obtained when S = 0. To account for the network topology, we assume that the capacities for processing flows P are different for different links, depending on the degree of nodes connecting this link. Then, Pi j = β(ki + k j ), where Pi j is the processing capacity of link i − j, ki is the degree of node i and β is the parameter. In given manual networks, starting from an unloaded network, the evolving mechanism of traffic flow can be described as: (1) Generation traffic flow. Assumed that the flows are generated at the nodes. At each time step, we impose a constant input of newly created traffic flow R. The source of each flow as well as its destination is chosen at random among all the nodes of the network. Besides, each node can send P flow which is related to the degree of the node at each time step and, as a consequence, one node, i, can have a queue to be delivered. (2) Movement. Flows move through the graph simultaneously searching for their respective destinations based on the shortest path algorithm. At each time step, only at most Pi j flows can be transported on link i − j according to the FIFO (First-In–First-Out) principle. When the queue at a selected link is full, the link won’t accept any more flows and the flow will wait for the next opportunity. Once a vehicle arrives at its destination, it will be removed from the system [14]. Because the link with the largest betweenness can be easily congested and the congestion can quickly spread to the entire network, it is necessary to consider only the traffic balance of this link. Since the flows are transmitted along the shortest paths from the sourcePto the destination, the probability that a created flow will pass through the link with the largest betweenness i is Bi / j B j [6]. At each time step, on average, R flows are generated. Thus, the average
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number of flows that the link with the largest betweenness receives at each time step is X Bj, F = R · N · hDi · Bmax /
(2)
j
where hDi is the average shortest path length of the network, N is the number of nodes in the network, and max is the index of the node with the largest betweenness [6]. We use the Eqs. (1) and (2) to derive the critical flow generation rate Rc . Let q A = vmax τ/2l0 ± (vmax τ/2l0 )2 − (d/l0 )2 , and C = L/l0 . Thus, F=
C−F , A
(3)
C . A+1
(4)
and, then, F=
Based on Eq. (1), the following equation holds: C Bmax = R · N · hDi · P . A+1 Bj
(5)
j
P
For the network, since j B j = R · N · hDi, thus we can calculate the critical flow generation rate Rc : P C· Bj Rc =
j
(A + 1) · N · hDi · Bmax
.
(6)
Up to now, we have obtained the critical value of the flow generation rate Rc , which can be used to describe the flow dynamics on networks. For a small value of the generation rate R, the flow assigned to the link is small so it can be processed in unit time. But for a larger R, the flow may exceed the link’s processing capacity, and it is more likely to accumulate at this link, resulting in traffic congestion. If the congestion cannot be alleviated, it breaks the equilibrium of the network and quickly spreads to the entire network. Therefore, it is important to determine the value of Rc so as to avoid and control the congestion effectively. 3. Simulation results Recently, there has been much research on the representation of urban networks [15–20]. After an in-depth investigation of both the topological (dual) graph [16], where streets are nodes and intersections are edges, and the spatial (primal) [17] graph, where intersections are turned into nodes and streets into edges, representations of street networks, Scellato et al. [18] have provided a tool for the analysis of the backbone of a complex urban system represented as a spatial (planar) graph. Kurant and Thiran [19] extracted the real physical topology from timetables. Jiang and Claramunt [20] proposed a topological analysis of large urban street networks based on a computational and functional graph representation. This representation gives a functional view in which vertices represent named streets and edges represent street intersections. While in this paper, links and intersections corresponds to the edges and nodes in the constructed urban traffic network. Generally, we represent networks as graphs G = (V, E), where V and E are the set of vertices and edges respectively. G is described by the adjacency matrix ei j . Define N as the size of the network. If there is an arc between node i and j, the entry ei j is the value 1; otherwise ei j = 0. In this paper, we use the Molloy–Reed algorithm [21] to generate scale-free networks. Selection of these particular structures is based on two reasons: (1) Scale-free is a most popular structure in the real world. A larger number of previous literature studies have concluded that most real networks display scale-free characteristics, including the Internet, WWW networks, power grids and so on [5,22,3];
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Fig. 1. The queue length S as a function of time steps for different flow generation rates R. The parameters are N = 600, L = 600, l0 = 8, d = 20, τ = 3, vmax = 20, β = 0.05.
(2) In our previous work [14], we find that the urban transit networks exhibit a power-law distribution with exponent λ ≈ 2.24. The congestion in scale-free networks is studied using computer simulation and the results are presented in this section. All results presented in this paper are for undirected, uncorrelated scale-free networks with an exponent of the power-law degree distribution λ ≈ 3 generated using the generation method mentioned above with hki ≈ 3. Using this exponent is based on the following reason: In classical references, it is obtained that most real-world networks follow power laws with exponent range 2–3. Meanwhile, the seminal work by Barab´asi and Albert (BA model) has constructed a scale-free network with exponent 3. Therefore, carrying out our study with exponent 3 makes it more universal and easier to generalize the results. It is assumed that at each time step, 0.1 unit flow is created and there are 0.1 × N flows generated in the whole network, and the parameter β = 0.05 in our paper. The simulation process is based on the evolving mechanism in Section 2. Throughout the paper, for each network, more than 25 realizations are averaged. Fig. 1 shows the time evolution of the queue length of traffic congestion S for different critical flow generations rate R on the link with the maximum betweenness. We see that, for all cases considered, S is zero at the beginning, and it suddenly increases when time is larger than a critical value. We also observe that S will increase with R, which means that the addition of flow can cause possible congestion that is most likely to occur at heavier links. As a result, phase transition can be delayed in the sense that the network can be more tolerant to traffic congestion for smaller values of R. Clearly, in comparison with the small R, the heavy flow on a central road (the edge with the highest betweenness) is difficult to process for larger R. Therefore, more and more traffic flow is accumulated and the traffic congestion occurs. In Fig. 2, we report the simulation results for S as a function of the maximum speed allowed, vmax . As one can c see, first, as vmax increases, S increases gradually; only when vmax is increased above a value vmax (in this paper, c c vmax ≈ 14) is a suddenly jump observed. This result suggests that, to a certain extent, decreasing vmax may relieve the traffic congestion. However, the larger of the maximum allowed speed vmax , the larger the headway is and the longer the time for the head vehicle to leave the queue. Thus, the queue length on links will increase. Fig. 3 shows the relationship between S and the total length of road L for the flow generation rate R = 0.1. The monotonic increase of S can be well approximated by the linear relation S ≈ 0.023L + 0.0028. This means that for a longer road, more drivers alternate it as the better trip route. As a result, this road is easy to be distributed flows and will become congested. Next, we compute the critical flow generation rate Rc and plot the relationships between Rc and τ , Rc and vmax in Figs. 4 and 5 respectively. The common characteristics of these two figures are that the critical generation rate Rc is decreasing with the increase of network size N . These figures also show that parameters τ and vmax have the same effects on the falling shapes of Rc for different network sizes N . However, the larger the network size is, the quicker
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Fig. 2. The queue length S as a function of the maximum allowed speed vmax with flow generation rate R = 0.1. Other parameters are N = 600, L = 600, l0 = 8, d = 20, τ = 3, β = 0.05.
Fig. 3. The queue length S as a function of the total road length L with flow generation rate R = 0.1. Other parameters are N = 600, l0 = 8, d = 20, τ = 3, vmax = 20, β = 0.05.
of Rc decreases, which means that for a larger traffic network, by only controlling microcosmic characteristics such as the speed, length of a single road, the passing capacity of the traffic network could not be improved well. In addition, Fig. 4 also shows that, clearly, for a smaller τ , the network size has great effects on Rc . But for a larger τ , Rc will approach a constant value and become independent of τ . This means that, the longer the time that the head car needed from congestion to the free state, the more serious the traffic congestion will be. In Fig. 5, we show Rc as a function of the maximum speed vmax . Our results suggest that the critical generation rate decreases with increase of vmax , which is consistent with the result drawn from Fig. 2. 4. Conclusion In conclusion, in this paper, by taking the microscopic characteristics on a single link into account, we derive the critical flow generation rate in terms of the whole network behaviors. Simulation results indicate that some values of microscopic characteristics on links have large effects on the queue length. And the critical flow generation rate decreases with increase of the network size N , maximum speed vmax and parameter τ . We find that considering only
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Fig. 4. The relationship between τ and critical flow generation rate Rc for different network sizes N = 200, 400, 600. Other parameters are L = 600, l0 = 8, d = 20, vmax = 20, β = 0.05.
Fig. 5. The relationship between the maximum allowed speed vmax and critical flow generation rate Rc for different network sizes N = 200, 400, 600. Other parameters are L = 600, l0 = 8, d = 20, τ = 3, β = 0.05.
the local information for a single link in a transportation design could not alleviate traffic congestion effectively. Thus, behaviors of the whole network must be analyzed simultaneously in order to improve the transport environment. Acknowledgements This paper was partly supported by the National Basic Research Program of China (2006CB705500), NSFC of China (70631001), Foundation for the Author of National Excellent Doctoral Dissertation of China (200763) and Changjiang Scholars and Innovative Research Team in University (IRT0605). References [1] [2] [3] [4] [5]
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