For bus stop network, the bus stops are chosen as nodes and an edge exists between two nodes if they are consecutive stops on the any bus line.^[15] In this way, the node degree k in bus stop network means the number of choices passengers can take to the consecutive stops and the distance d equals the total number of stops passengers need to pass on the shortest path from the start stop to the end stop, but sometimes it ignores the transfer times between two stops. Therefore, the bus stop network can intuitively represent the topology of the urban bus network.

By MATLAB software, we choose the Shenyang bus network as an example, the degree scope of bus stop network is in a range of [2, 13], the average degree of the whole network is 2.7894, and the degrees of most nodes are 2 or 3 since most of stops are in one line and they only connect the previous-stop and the next-stop. A few stops with larger degrees in Shenyang bus stop network are shown in Table 1. From Table 1, we can see that the stop with the largest degree is Shenyang Railway Station which connects 13 other consecutive stops.

Table 1.

Table 1.

Table 1. Top 5 stops with the larger degrees in Shenyang bus stop network. . Bus stops Shenyang Railway Station Shengjing Hospital Huigong Square Shenyang North Railway Station Environmental Protection Bureau Degree 13 12 11 11 11

Table 1. Top 5 stops with the larger degrees in Shenyang bus stop network. .

The distribution of degrees, clustering coefficient and distance of bus stop network are shown in Fig. 1. Figure 1(a) shows that the degree distribution of Shenyang bus stop network obeys power-law distribution P(k) = 4.469·k^(−2.894) and the degrees of most nodes are between 2 to 6. So the bus stop network of this city is a scale-free network.

Fig. 1.

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	Fig. 1. Characteristics of Shenyang bus stop network, (a) degree distribution of network, (b) distance distribution of network, (c) clustering coefficient distribution of network.

From Fig. 1(b), the distance d between any nodes ranges from 1 to 78 and the mean path length L is 22.0931. It denotes that reaching one stop from another stop would take an average of 22 stops for passengers in Shenyang bus network. From the studies of other cities,^[18,21–24] we find the mean path length L in Shenyang is longer than in Beijing (15), Taiyuan (14), Hangzhou (10), and Shanghai (8). This means that passengers will usually spend more time by bus in Shenyang.

From Fig. 1(c), we find the average clustering coefficient of Shenyang stop network is 0.0692, which is smaller than that of Beijing (0.158). Bigger clustering coefficient means that there are more connections between the nearest neighbor stops, which gives more alternative choices for passengers. So it means that it is easier to be congested and the robustness is worse in Shenyang.^[36]

The least transfer time means more convenient to travel by bus, so the transfer time is also an important factor for bus network. In bus transfer network, bus stops are also defined as nodes, however, an edge between two nodes exits if there is a direct bus line without transfer between two bus stops. In other words, if bus line A consists of nodes a_i, i.e., A = {a₁,a₂,...,a_n} then in the bus transfer network the nearest neighbors of the node a_i are the rest of the nodes in A.^[15] Consequently, the node degree k in bus transfer network is the total number of stops which are reachable without transfer from this node. For some bus stops which are contained in only one bus line, k + 1 is the number of stops in this line. d is the distance between two nodes and d − 1 equals the number of transfer times from one stop to another stop. Therefore, bus transfer network can represent the actual situation about transfer between bus stops.

The degree of Shenyang bus transfer network is from 2 to 503, the average degree of the whole network is 64.95, and the degrees of the most nodes are in a range from 20 to 61. This means that most of stops in Shenyang can reach other 20 to 61 stops without transfer. A few stops with larger degrees are shown in Table 2. The node with the largest degree is Shenyang North Railway Station, which can reach other 503 stops without transfer. Larger degree means more opportunities for the passengers to reach their destinations without transfer. Meanwhile, we find that the nodes with large degrees in bus transfer network also have large degrees in bus stop network, and these nodes can be denoted as the hub stops which can carry more traffic, so the robustness analysis based on these stops is meaningful.

Table 2.

Table 2.

Table 2. Top 5 stops with the larger degrees in Shenyang bus transfer network. . Bus stops Shenyang North Railway Station Wuai Market Station Shenyang Railway Xiaojin Bridge Square King’s Temple Degree 503 460 448 438 413

Table 2. Top 5 stops with the larger degrees in Shenyang bus transfer network. .

The distribution of degree, clustering coefficient and distance of bus transfer network are shown in Fig. 2. From Figs. 2(a) and 2(b), the degree distribution of Shenyang bus transfer network is close to the power-law distribution and distance d between nodes is in a range from 1 to 5. Mean path length L in bus transfer network reflects the average transfer time between bus stops. Larger mean path length means more transfer times for passengers by bus. After calculating, we obtain L to be 2.527 in bus transfer network, this means from one stop to another stop passengers will transfer more than one time in average, and it is not convenient for bus travel.

Fig. 2.

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	Fig. 2. Characteristics of Shenyang bus transfer network: (a) degree distribution of network, (b) distance distribution of network, and (c) clustering coefficient distribution of network.

In addition, from Fig. 2(c), we find that the clustering coefficients of most nodes are all 1 because these nodes are in a globally coupled network of one bus line. The average clustering coefficient is 0.741 which is the possibility with which the neighbor stops are in the same bus line. Larger clustering coefficient in bus transfer network means that bus lines link with each other more closely and transfer is more convenient. According to the small mean path length and large clustering coefficient, we can find that Shenyang bus transfer network is a small-world network. In general, the capacity of bus transfer in Shenyang is at a medium level and still needs to be developed.

Usually, passengers may take more than one bus to their destinations, so it is necessary to find the relations between bus lines. For bus line network, we define bus lines as nodes, that is very different from the bus stop network and bus transfer network, and if two different bus lines (nodes) contain one or more the same bus stops, we define that there is an edge between the two lines (nodes), in fact, these bus stops are transfer stops. The node degree k in this topology is the number of the other bus lines to which passengers can transfer. And the distance d can be explained as the number of transfer times through one line to another. Therefore, bus line network can describe the bus transfer between different bus lines.

Also, we take the Shenyang bus network for example, and there are 192 nodes and 5049 edges in bus line network. The degree of this network is in a range of [1, 105], the average degree of the whole network is 52.594, which means that one bus line may intersect with other 52 lines on an average. The top five lines with larger degrees are shown in Table 3, and it is easier for passengers to transfer by these bus lines. The nodes with largest degree are line No. 117 and line No. 152, which shares 105 stops with other lines.

Table 3.

Table 3.

Table 3. Top 5 bus lines with the larger degrees in Shenyang bus line network. . Bus lines No. 117 No. 152 No. 207 No. 134 No. 328 Degree 105 105 96 92 90

Table 3. Top 5 bus lines with the larger degrees in Shenyang bus line network. .

The distribution of degree, clustering coefficient and distance of bus line network are shown in Fig. 3. From the uniform distribution over a wide range shown in Fig. 3(a), we can find that the Shenyang bus line network does not have a scale-free property at all. In a bus line network, smaller mean path length means less transfer time between bus lines. As shown in Fig. 3(b), most of distances between two lines are in a range of [1, 3] and we can obtain mean path length L to be 1.791, which means that the passengers who should travel between lines need to transfer two or more times on an average. Compared with the scenarios in Hangzhou (0.89), Taiyuan (0.68), and Xi’an (0.44),^[18,21–24] transfer times in Shenyang is large, which means that the passengers usually need more transfers between bus lines in Shenyang. According to the small mean path length and large clustering coefficient, Shenyang bus line network is a small-world network.

Fig. 3.

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	Fig. 3. Characteristics of Shenyang bus line network: (a) degree distribution of network, (b) distance distribution of network, and (c) clustering coefficient distribution of network.

Clustering coefficient in bus line network determines the density of the same bus stops shared by the neighbor bus lines. As shown in Fig. 3(c), the average clustering coefficient is 0.537. Compared with the cases in Xi’an (0.71), Shanghai (0.56), and Beijing (0.54),^[18,21–24] the same bus stops between bus lines in Shenyang are sparse. Some main routes have much more bus lines while few remote routes have only one line.

We choose selective attack and random attack for invulnerability analysis. For the selective attack strategy, we remove some nodes with higher degrees according to their degree order from high to low. For the random attack strategy, the nodes are removed randomly. The main evaluation indexes are defined as follows.^[25,37,38]

From the above analyses based on both single layer and layered network, we find that for any model, the selective attack always causes more serious consequence than the random attack. Because the selective attack destroys the hub stops or lines, the hub stops or lines can affect the invulnerability of the bus network largely. So the managers should keep the hub bus stops or bus lines work unblocked and effective by reducing some project constructions there.

In addition, the comparison of the bus stop network with bus line network shows that the bus stop network is less robust and will affect the bus lines much more, that is to say, the break-down of bus stops may cause more serious result than that of the bus lines. So managers should pay more attention to the connectivity of bus stops than that of the bus lines

For the bus lines, some suburb area usually have only one bus line which may be isolated easily, therefore, for the bus network planning, bus lines in the suburb area need to be strengthened.

Multilayer structure network is well used in many fields. For instance, Each WWW or P2P link is mapped on the underlying IP network,^[40] each complete logistic process is determined by the underlying delivery site. So, all the three models above are only a part of large urban bus systems.

In this paper, urban bus network will be viewed as a two-layer network, one layer represents the physical infrastructures while the other layer represents the logical layer, i.e., the passenger flows on the infrastructures. That is, the nodes in the physical layer are the bus stops and the edges are the bus lines connecting neighbor stops. The node in the logical layer is also the stop while the edge is the trip demand which connects the first and the last stop of a particular trip.

Considering the definitions of bus stop network, bus transfer network and bus line network above, the bus transfer network describes a kind of passenger flow while bus stop network represents the information about the bus lines, which is the physical infrastructure of the flow in fact.^[41] So the layered bus network is composed of the bus stop network and the bus transfer network, which represents the physical layer G^φ = (V^φ, E^φ) and the logical layer G^λ = (V^λ, E^λ) respectively.

Because the logical layer represents the trip demand, we consider the logical layer as an undirected graph without the loads nor the capacity. And both layers have the same set of nodes (i.e. bus stops). Edge of logical layer is weighted by assigning it to physical layer, which is denoted by w(·) with w = 1. Each logical edge e^λ = (u^λ, v^λ) is mapped on the physical graph as a physical path M(e^λ) ⊂ G^ϕ connecting the nodes u^φ and v^φ, correspondingly the u^λ and v^λ in the logical layer.^[40] This path is based on the least transfer times and the least stops passing from start stop to end stop and if there are several equal choices, we randomly choose one of the paths. The set of paths corresponding to all logical edges is called mapping M(E^λ) from the logical topology to the physical topology. So now the load L(e^φ) of the edges e^φ in the physical layer is the sum of weights of all logical edges whose paths traverse this edge, and given as

Load on an edge in physical layer can be seen as the passenger flows traverse over it.^[35] Considering economic limitation, capacities of routes are also considered. Following Motter–Lai model,^[30] the capacities of an edge C_ij in physical layer are defined as

In the physical layer as shown in Fig. 9, if edge e_ij is overloaded and cannot provide the bus transit for more passengers, we say it as being broken down. Edges traversing over e_ij are all affected in logical layer. So passengers begin to change their paths, affected edge e^λ is deleted when there is no path in the physical layer, G^φ, otherwise, the logical edge e^λ remains in the network, and its mapping is updated by the shortest path in G^φ.

Fig. 9.

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	Fig. 9. Illustrations of the method of redistribution, showing (a) effect propagation in two-layer system when eij is broken down, and (b) dashed lines are valid in logical layer under rerouting.

Once an edge is broken in the physical layer, loads on this edge should be redistributed and some other edges will receive additional loads. If the total loads on other edges do not exceed their capacities, the cascading failures cannot be triggered. But if loads on the broken edge are high, after redistribution of the loads, some of the other edges are overloaded and then will be broken down. It will result in the further distribution of loads and may trigger cascading failures of the whole network finally. In the urban bus network, higer load means more passenger flows on infrastructures and edges with higher loads are the key routes in physical layer. The key edge is more representative and can influence the cascade failure progress largely.^[35,38] So in this paper, we only remove several edges with higher loads in the physical layer.

Here, we choose 4 parameters mentioned above, which are the relative size S, diameter D, mean path length L in the giant component, and network performance parameter E. At first, we remove top 10 edges with the highest loads. The results of the attack on the two-layer Shenyang bus network are shown in Fig. 10.

Fig. 10.

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	Fig. 10. Attack results in physical layer (a) relative size S, (b) diameter D, (c) mean path length L, (d) network performance parameter E versus tolerance parameter α.

In Fig. 10, horizontal axis represents the tolerance parameter α. Obviously, the higher the value of α for an edge, the larger capacities the edge has and the more difficult the edge to be broken down is, which means that each edge has enough capacity to deal with the additional loads. When α increases from 0 to 1, L monotonically decreases, which means that larger α makes the network more robust.

But considering S, D, and E, the curves are more oscillatory than L. Furthermore, from Fig. 10(a) and 10(b), we can see that the curves are not monotonic obviously. In fact, the curve in Fig. 10(d) is also not monotonic, for example, when α = 0.8, E is larger than when α = 0.9. So sometimes smaller α may increase robustness of the network. This is because when some edges are broken down in the physical network, the edges in logical layer cannot be mapped on the physical graph, the loads of this path cannot be redistributed to other edges, and for some of other edges in logical layer, they may change their paths so that the loads can be transferred to other uncongested physical edges. We call these broken edges critical edges. With the capacities decreasing, edges are easier to destroy, once the critical edges are broken down, loads of some nodes in network may reduce (We call the phenomenon “load-reducing phenomenon”). That means that in an actual urban bus network, if some of the congested edges between bus stops are closed, which causes passengers unable to make a detour then choose to go out later or change to other lower loads edges, traffic congestion will decrease.

In Fig. 11, relative size S of the giant component in logical layer is the same as the one in physical layer, which means that some stops are not reachable due to the cascade failures. But considering L, D, and E, more load-reducing phenomena emerge. Whenα = 0.8, mean path length L is smaller than with α = 0.9, network performance parameter E is larger than with α = 0.9, that means more paths are broken down when α = 0.9 in logical layer (more broken paths cannot be replaced). When α = 0.4, diameter D in giant component is larger than with α = 0.3. So this directly confirms our explanation about the load-reducing phenomenon.

Fig. 11.

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	Fig. 11. Attack results in logical layer, showing (a) relative size S, (b) diameter D, (c) mean path length L, and (d) network performance parameter E versus tolerance parameter α.

In this subsection, we consider the passenger flows and analyse the dynamic cascade failures in the urban bus network. As a result, “load-reducing phenomenon” is found when some of the critical edges are broken down. According to this conclusion, when the congestion occurs, the managers may close some of the congested bus lines ephemerally between some bus stops (such as by changing the routes of the bus lines or taking no passengers at these bus stops), so the passengers will choose to go out later or change to other path which will reduce the loads in the congested lines. In fact, it is like the way of “odd-and-even license plate rule” in some big cities of China.