Based on the research progress in related fields and the distribution characteristics of road networks in valley cities, the complex network model of a city road network is established to study its connectivity reliability. Taking Lanzhou as the example, several parameters of the complex network abstracted from the road network are calculated and the practical meanings of them are described, respectively. On this basis, through computing the global efficiency and the relative size of the largest connecting subgraph under intentional attacks and random attacks, respectively, the curves of the above two parameters varying with the attacking times are drawn. The detailed investigation of connectivity reliability of Lanzhou road network is done by analyzing the curves’ tendency. Finally, we find that the network of a valley city has a poor connection and has a lot of dead ends. Besides, the average length of the roads is very long and the holistic connectivity reliability is at a lower level; these are suitable to the group-type distribution of valley city’s road network, and the connectivity reliability of the road network is stronger under random attacks than that under intentional attacks.
Since the reform and opening up of China, the urbanization process has been accelerated greatly, which requires the urban road system to meet the requirements of the rapid development of cities. However, due to the limitations of historical and natural conditions, in many cities, especially the west valley cities, the phenomenon of urban road construction not meeting the requirements of urban development has appeared, which has brought inconveniences to the people's production and life travel and has led to a series of social problems. Lanzhou, as the transportation hub and logistic center in northwest China, is suffering from particularly serious transport problems. Study on the reliability of urban road network will be able to provide theoretical supports for pertinently changing city traffic congestion.
Complexity science is the further development, enrichment, and deepening of systematic science and nonlinear science, and is the latest and the most forward area of scientific research. The complex network method, developed in recent years, has provided a new perspective for the research of the complexity of a system. Two articles [
Connectivity was first proposed by Mine and Kawai in 1982, which reflects the probability to maintain connectivity between nodes in transport networks. Asakura puts forward the concept of travel time reliability in 1991, which is another measure method of network reliability, fully taking into account the travel needs of the road network and traveler behavior [
Through the analysis of the literature, we can see that the study of reliability of road network has transformed from the study only considering the physical structure of the road network to that of consideration of loading traffic flow and the mutual influence between travel demand and the capacity of road network, as well as the travel behavior of road users; the corresponding reliability indexes are connectivity reliability and travel time reliability. During this period, some scholars have proposed demand satisfaction reliability, the weak point reliability, performance index reliability, and suffered reliability. In China, research is conducted on the basis of foreign studies, and is the supplement or continuation of foreign study results; there is no significant breakthrough in the road network reliability indexes, and most of the researches are on connectivity reliability and travel time reliability. But domestic researches have explored the connectivity reliability and travel time reliability from different perspectives and conditions, which widen the thoughts and provide a foundation for further study on this area. There are also scholars that have studied the road network capacity reliability.
It can be seen from the results in the field that research on transport network reliability is increasing every year, which reflects the increasing travel demand in daily lives and higher requirements proposed by travelers on the travel system. In this paper, we studied the connectivity reliability of the valley city road networks using the theory of complex network.
From the perspective of geography, considering the location of the city and the relationship between the city and the valley, we define the valley city as follows: a city with the urban built-up area or the main or core part of the built-up area located in a valley, and the development of the main body of the city is strongly directly restricted by the valley terrain and the surrounding mountains or hills [
Compared with a plain city, the valley city road network presents its unique features. Firstly, with the limitation of the mountains and rivers, and in order to reduce the slope of the road, the valley city road lines are often built along the mountains or rivers, and road network form is always the freestyle. The valley city road lines are adapted to the natural terrain in order to save road engineering cost. Most of the road is not a straight line, and the road network is irregular. The valley city road network makes it difficult to form the ring road. Another big characteristic of a valley city is that the transportation is mainly concentrated on the main road of the urban axis. With the influence of the banded terrain and the problems of shortage of land, there is not enough land space to expand the urban road and share the traffic pressure when the urban trunk road is saturated. To research the connectivity reliability of valley city road network we will provide the reference to urban road network design and construction.
The complexity network can be described accurately using the language and symbols of the graph theory. The findings and techniques in graph theory about network have been widely transplanted into complexity networks. Based on this, the urban road network model has been established based on complexity network.
For the urban road network, intersections of the city's road network can be abstracted as nodes, and road sections connecting the intersections can be abstracted as edges in the network. Therefore, the city's road network can be abstracted as the complex network composed by intersection and road sections connecting the intersections.
The following assumptions have been made when constructing the complex network to simplify the issue.
(1) If from intersection A in the urban road network, we can get to intersection B through a certain road section, and through the same road section, we can get to intersection A from intersection B, we consider the above connections as the same one, namely, implementing undirected treatment on the network.
(2) Ignore the actual length of the road sections that connect nodes of the urban road network. Assume that length of road sections are all 1, then the distance between nodes can be expressed as the number of road sections between the nodes, which do not consider the weights of the connections, and abstract the urban road network as a nonweighted network.
(3) In this work, residential districts at road sections, river crossing transport facilities, and ends of roads are considered as network nodes, which are equivalent to intersections. At the same time, if there is more than one road section connecting two nodes, then consider these sections as one connection, which is only one side between the two nodes. Consider road sections connecting two intersections directly in a reality network as sides with length = 1 in the model; from any intersection in the network, we can get to any other intersection, namely, the entire network is connected.
Based on the definition of the system reliability and the temporal and spatial imbalance of the urban transport travel, this paper presents the general definition of the urban road network reliability as: the statistical probability for the urban road network meeting a particular traffic travel demand under different spatial and temporal distribution.
Urban road network connectivity reliability based on the complex network is defined as follows: the ability for urban road network to maintain the connectivity state, after being attacked in different ways and the network suffering from a certain level of destruction.
The degree
In complex networks, the distance
The path length of urban road network is the distance from one intersection to another and the average path length
The clustering coefficient is the characteristic parameter used to describe the tightness of the network, marked with
The entire network's clustering coefficient
The clustering coefficient of the urban road network reflects the aggregation of an intersection and its
The efficiency of the network is qualified to analyze the network of small-world behavior instead of the average path length and clustering coefficient. The efficiency
Taking into account the situation of nonconnected graph, the local features of the network can be represented by the average efficiency of the partial subgraph
For the urban road network,
The largest connected subgraph of the network means the subgraph connecting all nodes with least sides. The relative size
The relative size of the largest connected subgraph weighs the capacity maintaining the original function of a network after destruction.
In the study, the existing nodes and road sections of Lanzhou city were selected (a total of 349 selected intersections and 507 road sections). Calculating the degree and degree distribution of Lanzhou city road network, the average path length and the clustering coefficient is helpful to studying the topological properties of the Lanzhou city road network. Analyzing these statistical eigenvalues has a practical significance for learning the valley city road network and its connectivity reliability.
For the Lanzhou city road network model, node degree reflects the importance of the node in the network; its practical significance is the number of road sections connected to the intersection in different directions. Figure
Degree distribution of Lanzhou City road network.
It can be seen from Figure
Nodes with a degree = 3 in the Lanzhou city road network account for more than 40% of the total nodes, indicating that junctions of three roads account for a large proportion in the Lanzhou city road network, this is because of the limitations of topographical, the trunk line distribution of Lanzhou City is usually along the rivers and mountains, and the secondary roads, slip roads connected to the trunk roads, which is very easy to form junctions of three roads. Nodes with a degree = 4 account for the second largest proportion. No matter what type of urban road network, common road intersections are the convergence of four directions, which meets the actual situation; the grid distribution and the irrational square format local road network in Lanzhou result in a lot of crossroads, whose corresponding node degree value is 4. Nodes with degrees = 1, 2 account for about 20% of the total nodes, and the number of nodes with a degree = 1 is more than that with a degree = 2, which equates with the situations of Lanzhou. Because the road network is not perfect in Lanzhou, there are a lot of cul-de-sac with a corresponding degree = 1; in addition, city roads in Lanzhou show long strip distributions, and there are a lot of living areas distributed along the strip roads; the intersections of the living areas and the roads are considered as nodes with a degree = 2. The above analysis shows that the model results are consistent with the actual situation.
The path length of the network represents the minimum number of road sections from a node to the specified node. Corresponding to the urban road network in Lanzhou, its practical significance is the number of road sections from one intersection to the specified intersection. Figure
Statistical chart of the average path length of Lanzhou.
Another important statistical characteristic parameter of complex network is the clustering coefficient, which is a physical quantity measuring the density of the network between adjacent nodes. The practical significance of the clustering coefficient corresponding to the road network of Lanzhou is the aggregation level of intersections in Lanzhou. Figure
Statistical chart of the clustering coefficient of Lanzhou.
Complex networks usually face two kinds of attacks: random attacks and deliberate attacks. As a small world network, the urban road network in Lanzhou is also facing these two attacks. This paper studies the changes of the connectivity reliability of Lanzhou road network under the two attacks. According to the actual situation of the urban road network in Lanzhou, the following specific assumptions were made for the two attack patterns.
The random attack refers to nodes in the network being deleted randomly at a certain probability. Taking into account the relatively large number of intersections, each attack will remove 5 nodes in the network, corresponding to 5 intersections of the urban road network in Lanzhou City.
The deliberate attack is the other attack mode the complex network suffered, which is a strategic attack on the nodes in the network. The purpose of this attack is to make the most severe damage with the least number of attacks. Therefore, this attack aims at and destroys the node with the greatest degree value of the network. In this work, the purpose of the deliberate attack is to make the most severe damage and make the road network of Lanzhou collapse as soon as possible. To make contrast with the random attack, 5 nodes with the greatest degree value will be removed from the network, corresponding to the 5 intersections with the most convergence.
Based on the above assumptions, the global efficiency
(1) Change curves of global efficiency
The global efficiency is an important indicator measuring the overall road network connectivity of Lanzhou. The calculated Lanzhou road network global efficiency is 0.118, indicating that the connectivity reliability of Lanzhou is poor.
Figure
Change curves of global efficiency under different attacks.
Under deliberate attacks, the global efficiency will decrease to below 0.06, which is only half of the original value, after 11 attacks (the deleted nodes only accounting for 15.7% of the total nodes); the network global efficiency will decrease to 0.02 after 23 attacks (deleted nodes accounting for 33% of the total nodes), indicating that the connectivity of the network reliability has become very poor; after 45 attacks (the deleted nodes accounting for 64.5% of the total nodes), the global efficiency of the network will become 0.00013 (close to 0), which shows the road network has been near-collapse. Under random attacks, after 14 attacks (the deleted nodes only accounting for 20% of the total nodes), the global efficiency will decrease to below 0.06; after 35 attacks (the deleted nodes accounting for 50.1% of the total nodes), the global efficiency will decrease to 0.02, which means that the connectivity reliability will become poor when more than half of the nodes are attacked; after 45 attacks (the deleted node accounting for 64.5% of the total nodes), the global efficiency becomes 0.01279, which is much larger than that of the deliberate attack (0.00013), and until this time, the Lanzhou road network has not been crashed.
The reason for these results is that there are great differences in the deletion of nodes under the two attacks. The deliberate attack is carried out in accordance with the degree value of the nodes. After the nodes with relatively larger degree value of the network, which play a very important role in maintaining the road network connectivity reliability, are removed, dramatic changes will occur on the topology fabric of the road network, leading to a great number of isolated intersections, and ultimately the collapse of the entire network. However, for random attacks, the probability for removing nodes according to the order of deliberate attacks is very small, and the network will demonstrate stronger robustness than that of deliberate attacks when subjected to random attacks, so more isolated intersections are needed to make the same extent of damage as the random attack. The above analysis shows that the damage on the connectivity reliability of the Lanzhou road network subjected to random attacks is slighter than that of deliberate attacks.
(2) Change curves of the relative size
Figure
Change curves of relative size of the largest connected subgraph under different attacks.
Under deliberate attacks, the relative size
The reason for these results is that when the network is subjected to deliberate attacks, the node removing is carried out in accordance with the degree value of the nodes; after the nodes with relatively larger degree value are removed during the early attacks, the network will be instantly differentiated into several subnets with sharply reduced nodes, so the relative size of the largest connected subgraph will change dramatically, the result of which is the maintenance of the degree of nodes left in the subgraphs at a very low level. When the network is subjected to random attacks, the deleting order is completely random, and the node with the largest degree value may remain in the largest connected subgraph until the end of the 45th attack; so in this case, the relative size of the largest connected subgraph may still have a relatively larger value compared with deliberate attacks. The above analysis shows that the connectivity reliability of the Lanzhou road network subjected to random attacks is better than that of deliberate attacks.
The paper constructs a complex network model of a valley city road network. Taking Lanzhou as an example of a typical valley city, the connectivity reliability of the road network was researched and the following conclusions are obtained.
(1) Valley city road network has poor connection and a lot of dead ends, and the number of road sections between intersections is too large, which makes it a poorly compacted network.
(2) Valley urban road network has long average path length and poor overall connectivity reliability, consistent with its cluster long strip layout; and the connectivity reliability of the urban road network subjected to random attacks is better than that of deliberate attacks, which is consistent with the characteristics of complex networks under attack.
This work is partly supported by the Humanities Social Sciences Programming Project of the Ministry of Education of China (no. 10YJA630126) and the State Social Science Fund Project (no. 11CJY067).