Exploring the Cascading Failure in Taxi Transportation Networks

Abstract

To explore the ability of taxi transportation service capacity in unexpected conditions, based on the taxi GPS trajectory data, this paper presented a taxi transportation network and explored a cascading failure model with the non-linear function of traffic intensity as the initial load. Moreover, the cascading failure conditions for different initial loads with different parameter settings were derived by combining the complex network theory. We verified the ability of taxi transportation networks to withstand unexpected conditions and analyzed the differences and features of taxi transportation service capacity for different areas of Lanzhou city. Three sets of comparative simulation experiments were implemented. The results show that when the initial load regulation factor $\alpha <1/\theta$, the failure of nodes with smaller initial loads in the network is more likely to cause cascading failure phenomena. When $\alpha >1/\theta$, the failure of nodes with larger initial loads in the network is more likely to cause cascading failure phenomena. Additionally, when $\alpha =1/\theta$, there is no significant correlation between whether cascading failure phenomena occur in the network and node loads. This study can provide a prior basis for decision-making in the management of urban taxi operations under different passenger flow intensities.

1. Introduction

As an essential part of the urban public transportation system, taxis play an indispensable role in ensuring the travel of urban residents. However, in the event of unforeseen events such as taxi charging station failure, traffic congestion, the surge in passenger flow, epidemic blockade, etc., a reduction in taxi service capacity in one area of a city or a surge of travel demand within a short time will affect the efficiency of the entire city taxi transportation system.

From the perspective of complex networks, the network system often has a large amount of load traffic. When one or more nodes in the network fail due to external effects, the related loads will be evacuated along the edges connected to it, which then redistributes the load traffic on the network. When the newly allocated load exceeds its capacity limit, a new round of node failures will be triggered, eventually leading to the failure of large-scale nodes in the network. This phenomenon is called cascading failure [1,2]. Scholars have mainly studied the cascading failure problem in computer networks [3] or power networks [4,5]. Nowadays, the main cascade failure models are Sandpile model [6], load-capacity model [7], coupled map lattice model (CML) [8], KQ-cascade model [9,10], etc. Among them, the load capacity cascading failure model proposed by Motter [7], which fits best with the mechanism of traffic congestion, has been widely used to study the invulnerability or robustness of a single-layer public transportation network. Many studies have been carried on capacity control [11,12], load distribution [13,14], node state [15,16,17], and other aspects in many fields, such as road transportation [18,19], rail transport [20,21,22], regular bus [23], urban agglomeration transportation network [24], and other network scenarios [25].

For the cascade failure of conventional rail transportation or bus transportation, the failure of a node represents the interruption of a certain line or the damage of a certain station, but for the taxi transportation networks without fixed routes and stations, a node of the network refers to a certain area of the city. The failure of a node means that it is difficult to take a taxi in a certain area for a long time. This regional lack of passenger service capacity must have some impact on taxi operations in other regions. Therefore, studying the cascade failure of the taxi transport networks can better identify the differences and variations in service capabilities across regions.

Many scholars regarded the betweenness as the initial load of the node in the related research. The network topology significance is expressed by the betweenness in most networks, which is consistent with the routing decision of the traveler. However, there are also some real network node flows that do not have clear regularity. To address this issue, some scholars improved the models related to the initial load, such as setting the initial load as a linear or non-linear relationship between betweenness or degree functions [26,27,28,29]. These improved models promote the effectiveness of the cascading failure model to describe the survivability of real complex systems, but they focus on theoretical analysis, and most of the simulation processes are still based on scale-free networks, which lack empirical evidence of real-world networks. Furthermore, taxis are widely distributed in cities with autonomous routing compared to modeling approaches for rail transit or bus networks such as L-space, P-space, and C-space [1,30,31]. The taxi transportation network has high traffic density, high complexity, and strong time-varying ability. Obviously, all the pick-up and drop-off stations cannot be exhaustively listed in the modeling process. Therefore, limited by the abstract nature of the taxi transportation networks, there are few studies on the cascading failure of the taxi transportation networks.

In this paper, we used the Lanzhou taxi GPS trajectory data to extract the OD (Origin-Destination) information of trips, and the K-means ++ clustering algorithm is used to divide the travel information into taxi-operating districts for transforming “regions” into “nodes”. We constructed a directional weighted network of taxi passenger flow considering the quantitative relationship between node traffic intensity and node degree in the taxi transportation network. Additionally, a cascade failure model for the actual passenger flow taxi transportation network is proposed. Finally, the evaluation for cascade failure is empirically analyzed through theoretical analysis and simulation experiments.

The remainder of this paper is organized as follows: Section 2 introduces a representation method of taxi transportation networks based on trajectory data and calculates the basic topological characteristics of the taxi transportation network in Lanzhou under this method. In Section 3, a cascading failure model with actual flow intensity as the load is proposed. In Section 4, the critical conditions for cascading failures of our model under different flow scenarios are derived. In Section 5, three sets of comparative simulation experiments are designed to verify the laws proposed in this paper. In Section 6, at the end of this paper, the conclusions and future studies are discussed.

2. Construction of Taxi Transportation Networks

2.1. Data Cleaning and Map Matching

More than 800 million GPS trajectory data is generated by about 10,000 taxis in Lanzhou during April 2022. The GPS trajectory data include a variety information, such as the taxi license plate number, coordinates, acquisition time, taxi operation status, etc., as shown in

Table 1. Taxi GPS Data Fields.

Field Name	Description	Example
vehnum	Taxi license plate number	GanA80001
longitude	GPS longitude	103.69802
latitude	GPS latitude	36.10697
gpstime	GPS time	12:46.0
taxistatus	Taxi operation status, 0 means empty, 1 means passenger	0
velocity	The velocity of the taxi currently	0

Note: The taxi operation status ignores the device delay in this paper, and regards the 1 point changing from 0 to 1 and the 0 point changing from 1 to 0 as the positions of passengers getting on and off.

.

Errors or deviations may inevitably occur due to unstable signals or equipment failures; therefore, cleaning duplicate abnormal data of the trajectory data is necessary. According to the vehicle status information, the taxi pick-up and drop-off data are extracted. A map-matching algorithm based on the road geometric relationship [32,33] is used to correct the location of the taxi pick-up and drop-off points. Finally, an average of 270,000 valid pick-up and drop-off point data per day are obtained.

2.2. Spatial Clustering and Network Construction

The K-means ++ algorithm [34] is used to obtain $k$ clusters, and the information of the drop-off points are counted in each cluster, which is the traffic connection between each sub-region $A_i$ and other sub-regions $A_j$. In this way, a taxi travel flow matrix is established:

$$OD=\left[\begin{array}{ccc}o{d}_{11}& \cdots & o{d}_{1k}\\ ⋮& ⋮& ⋮\\ o{d}_{k1}& \cdots & o{d}_{kk}\end{array}\right](i,j\in \left[1,k\right])$$

The cluster center is regarded as the center of gravity of the taxi operation area $A_i$, which is the node $v_i$ of the taxi transportation network. If there is a traffic connection between any two areas $A_i$ and $A_j$, a directed edge $e_{ij}$ will be generated between nodes $v_i$ and $v_j$, and the traffic flow between $A_i$ and $A_j$ is the edge weight $w_{ij}$, which is used to construct a directed weighted complex network for taxi transportation $\mathit{G}$, where

$$\begin{array}{c}G=\left\{V,E,W\right\}\hfill \\ V=\left\{v_1,v_2,v_3,\cdots ,v_k\right\}\hfill \\ E=\left\{e_{11},e_{12},e_{13},\cdots ,e_{kk}\right\}\hfill \\ W=\left\{w_{11},w_{12},w_{13},\cdots ,w_{kk}\right\}\hfill \end{array}$$

In this process, we first need to determine the value of $k$, the number of clusters, which in this paper determines the size of the urban space division. Regarding the scale of division, when the division scale is too large, the division cell is too coarse, which makes the analysis unable to be detailed, and the short-haul traffic within each cab operation cell is not counted in this case, which will lead to insufficient sample size of inter-regional traffic connections. When the scale of division is too small, the sample size in the division cell is small, the regional characteristics are diluted, and it is difficult to reflect the differences among regions, and it is easy to have too few samples in some regions, which will lead to the loss of statistical significance.

Most researchers in the field of transportation tend to determine the cluster size exclusively by referring to census tracts or the concept of “traffic cell” in traditional traffic engineering theory. The literature [35] used census tracts as a reference basis for division and constructed a Chicago taxi OD network with 679 nodes, and the literature [36] used traffic tracts in the Xi’an urban transportation plan as a reference basis for division and constructed a Xi’an city taxi OD network with 422 nodes. In order to find the appropriate division scale for the main city of Lanzhou, two indicators, intra-regional traffic share and average distance between neighboring regional centroids, are designed in this paper.

Intra-regional traffic share: the ratio of the sum of short haul traffics whose origins and destinations are all within the same region to the sum of total traffic.

$$\eta =\frac{{\displaystyle {\sum }_{i=j}o{d}_{ij}}}{{\displaystyle {\sum }_{i,j\in K}o{d}_{ij}}}$$

Average distance between neighboring region centroids: the actual distance is calculated between all region centroids two by two, and then the average of the top k distances is taken in order of smallest to largest.

$$\overline{D}=\frac{1}{k}\min \left\{{\displaystyle \sum _{i,j}^{k}D(C_i,C_j)}\right\}$$

where $D(C_i,C_j)=R·\arccos \left[\cos (y_i)\cos (y_j)\cos (x_i-x_j)+\sin (y_i)\sin (y_j)\right]$, $x_i$ and $y_i$ are the latitude and longitude coordinates of the center point $C_i$, and $R$ is the radius of the Earth, which is 6371.004.

As can be seen from Figure 1, at $k=400$, there is a clear “elbow phenomenon”, and the number of samples ignored at this time accounts for about 3 percent of all, which is appropriate. We can see in Figure 2 that after the value of $k$ exceeds 400, the average distance between the centers of each adjacent area is already less than 450 m, which means that many areas are already small. Considering the urban scale of Lanzhou main city and referring to the results of traffic cell division of Lanzhou city traffic planning, we can see that if the division is even smaller, this will cause some areas of the same nature to be wrongly cut apart. Therefore, the main city of Lanzhou is divided into 400 taxi operation areas. The visualization of the delineation results is shown in Figure 3. From a macroscopic point of view, we can assess whether the scale of the division is adapted to the scale of the city. We can see that in urban centers, where people are socially active, the study cells are divided into small and dense; whereas, in suburban areas the study cells become relatively large and sparse, which is determined by the density of taxi distribution in the city. This Figure demonstrates to some extent the advantage of clustering being more flexible and better able to capture key information than grid division.

Then, a taxi transportation network with 400 nodes and 25,571 non-zero weight edges is constructed. The approximate topological characteristics of the taxi transportation network can be seen in Figure 4. It is a tightly connected network structure, in which a large number of nodes near the center of the network are connected to each other, and only individual nodes on the periphery of the network have few neighbors, which implies that the taxi transportation network is characterized by a large amount of information, a high level of activity, and a wide range of connections. Figure 5 reflects the actual geographical location of each node of the network. It can roughly reflect the flow characteristics of cabs in different areas within the city: the areas in the center of the city have the most frequent and closest traffic connections with each other, all suburbs have a tendency to flow to the central area, but there are few traffic connections between suburbs and suburbs.

2.3. Analysis of Topological Characteristics of Taxi Transportation Networks

Assuming that the taxi models are consistent in a large-scale sample and that the single vehicle carries a stable number of passengers per trip, then the number of taxi passengers carried in each region can be considered as the taxi traffic intensity in each region, specifically, the traffic intensity at node $v_i$ is the sum of the edge weights associated with node $v_i$, then the traffic intensity of the nodes in the taxi OD network can be expressed as:

$$S_i={\displaystyle \sum _{v_j\in N_i}{w}_{ij}}$$

where $N_i$ is the set of neighbors of node $v_i$, and $w_{ij}$ is the traffic flow in the direction from node $v_i$ to node $v_j$.

The average path length $L$ is defined as the average of the distances between any two nodes,

$$L=\frac{2{\displaystyle \sum _{i\ge j}\mathrm{dis}(v_i,v_j)}}{N(N-1)}$$

where the distance $\mathrm{dis}(v_i,v_j)$ between nodes $v_i$ and $v_j$ is defined as the number of edges on the shortest path connecting these two nodes.

Since the taxi transportation network is a directional weighted network, according to the method given in the literature [37], the clustering coefficient is calculated as:

$$C_i^W=\frac{{(W^{\left[\frac{1}{3}\right]}+{(W^T)}^{\left[\frac{1}{3}\right]})}_{ii}^3}{2\left[d_i^{}(d_i^{}-1)-2d_i^{\leftrightarrow }\right]}$$

where $W^{\left[\frac{1}{3}\right]}=\left\{w_{ij}^{\frac{1}{3}}\right\}$, i.e., the matrix obtained from $W$ by taking the third root of each entry. $d_i^{\leftrightarrow }$ is the number of bilateral edges between $v_i$ and its neighbors (i.e., the number of nodes $v_j$ for which both an edge $e_i\to e_j$ and an edge $e_j\to e_i$ exist).

The average degree, average path length, aggregation coefficient, and other topological characteristic indicators of the taxi transportation network are calculated by the python networkx package, as shown in

Table 2. Relevant characteristic indicators of taxi transportation network.

Number of Nodes	Number of Edges	Average Degree	Average Traffic Intensity	Average Path Length	Clustering Coefficient
400	25,571	128	309	1.973	0.437

.

Figure 6 and Figure 7 show the order of node degree and node traffic intensity from small to large.

As shown in Figure 6 and Figure 7, the node degree and the traffic intensity conform to the power-law distribution. The closeness of transport links and the volume of passenger traffic varies greatly from region to region. In order to investigate the quantitative relationship between nodal degree and nodal intensity, we set three functions ($S(d)\approx A·d^{\theta }$, $S(d)\approx d^{\theta }$ and $S(d)\approx A·d\pm C$) and used Origin software to fit the nodal degree and nodal traffic intensity and calculate their residual sum of squares, R-squared (COD), and adjusted R-squared, according to the fitting results of different functions in Figure 8 and

Table 3. Fitting state statistical indicators.

Function	Fitting Results	Sample Size	Degree of Freedom	Residual Sum of Squares	R-Squared	Adjusted R-Squared
$S(d)\approx A·d^{\theta }$	$S(d)\approx 0.38·d^{1.37}$	400	398	3,922,132.89368	0.78279	0.78225
$S(d)\approx d^{\theta }$	$S(d)\approx d^{1.18}$	400	399	4,092,719.0344	0.77335	0.77335
$S(d)\approx A·d\pm C$	$S(d)\approx 2.94·d-66.46$	400	398	4,282,136.81186	0.76286	0.76226

, the nonlinear function has the best fitting effect. The fitting result is $S(d)\approx 0.38·d^{1.37}$.

3. Cascading Failure Model Construction

3.1. Initial Load and Capacity Control

Compared with topological indicators such as degree or betweenness, the passenger flow intensity of the network can more genuinely reflect the initial load of the network in the taxi transportation network. In order to better simulate the time-varying characteristics of the urban taxi transportation network and reflect the traffic state under different traffic flows, we propose to use the nonlinear function of the passenger flow intensity as the initial load of the node:

$$Q_i(0)=S_i^{\alpha }$$

(1)

Among them, $Q_i(0)$ refers to the initial load of node $v_i$, $\alpha >0$ refers to the initial load distribution coefficient, and $S_i>1$ refers to the network passenger flow intensity, then this is a monotonically increasing function of $\alpha$. In this paper, we use $\alpha$ to regulate the initial load size loaded to each node of the network, the larger $\alpha$ is, the greater the load on each node in the network we study, which implies a peak traffic scenario. Conversely, the smaller $\alpha$, the smaller the load on the nodes in our studied network, which implies a low traffic scenario.

It is assumed that the maximum number of taxi services within region $A_i$, which is also the maximum tolerable load of the node, is defined as the node capacity in this paper and is expressed as a linear function of the initial load:

$$C_i=\beta Q_i(0)$$

(2)

To ensure that no traffic overflow occurs in any node in the network, we set $\beta \ge 1$, the node backup capacity coefficient.

3.2. Node Status Recognition

After obtaining the load from the adjacent node $v_j$ at time $t$, the load of node $v_i$ at time $t$ is:

$$Q_i(t)=Q_i(t-1)+\Delta Q_j$$

(3)

The state of node $v_i$ at time $t$ is defined as:

$$is(v_i)=\{\begin{array}{c}1\\ 0\end{array} \begin{array}{c}{Q}_i(t)\le C_i\\ Q_i(t)>C_i\end{array}$$

(4)

Then, the total load transmitted at node $v_i$ to all its neighboring nodes at time $(t+1)$ is:

$$\Delta Q_i(t+1)=is(v_i)·Q_i(t)$$

(5)

When the traffic demand of region $A_i$ at the moment of $t$ is less than the taxi service capacity that the region can provide, it means that the load of node $v_i$ at the moment of $t$ is less than its capacity, that is, when $Q_i(t)\le C_i$, the node is in the normal state and does not transmit the load to the neighboring nodes; when the traffic demand of region $A_i$ at the moment of $t$ is greater than the taxi service capacity that the region can provide, it means that the region needs the taxis cruising in other regions at this time to come to help, and this process can be regarded as the traffic demand of the region is borne by the cabs in other regions. Expressed in the network is that the load of node $v_i$ at the moment of $t$ exceeds its capacity, specifically, when $Q_i(t)>C_i$, the node is in a failed state, and at the next moment, all its loads will be distributed to its adjacent nodes, and no new loads will be accepted.

3.3. Load Redistribution

There are two methods of load redistribution: one is network-wide distribution, the load of the failed node is redistributed to the entire network, and the other is local distribution, that is, the load of the failed node is distributed to its adjacent nodes according to specific routing rules. In the real taxi transportation scenario, if the supply and demand in region $A_i$ is out of balance, the taxis or travelers in this area always prioritize the adjacent area. Therefore, for the taxi transport network, the local load redistribution method is more effective, and it is often more inclined to select the node with a larger capacity. The load distributed by each adjacent node can be expressed as:

$$\Delta Q_j=\frac{C_j}{{\displaystyle {\sum }_{\phi \in \Omega }{C}_{\phi }}}{Q}_i$$

(6)

where $\Omega$ is the set of adjacent nodes of the failed node $v_i$.

3.4. Network Failure Evaluation Index

In this paper, the proportion of surviving nodes indicates the average network failure level. Specifically, it is the ratio of the nodes still surviving in the process of cascading failure to the total number of initial nodes:

$$R=\frac{N^{\prime }}{N}$$

(7)

3.5. Cascading Failure Simulation Process

Step 1: Build a taxi transportation network model and establish a region-passenger flow adjacency matrix.

Step 2: According to the area-passenger flow adjacency matrix, set the initial network load according to Formulas (1) and (2).

Step 3: Attack a node in the network to make it invalid.

Step 4: Evacuate the initial load of the failed node to adjacent nodes according to Formula (6).

Step 5: Update the load of each node in the network according to the Formula (3), judge the current state of the adjacent nodes according to the Formula (4), update the network information, and record the set of failed nodes and the set of surviving nodes at this time.

Step 6: After the adjacent node receives the load from the failed node, determine whether the cascading failure occurs according to Formula (4). If cascading failure occurs, start a new round of cascading failure simulation, and perform Step 4. If not, perform Step 7.

Step 7: Calculate the proportion of surviving nodes in the cascading failure process according to Formula (7).

End.

4. Cascading Failure Model Analysis

From the cascading failure model and simulation process, it can be known that the key condition to avoid the cascading failure at any time $t$ is:

$$Q_j(t-1)+\Delta Q_i\le C_j$$

(8)

Substitute Equations (1), (2), and (6) into:

$$Q_j(t-1)+\frac{C_j}{{\displaystyle {\sum }_{\phi \in \Omega }{C}_{\phi }}}{Q}_i\le \beta S_j^{\alpha }$$

(9)

For a taxi transportation network, a node that fails will transmit its load out, but if it does not fail, it will not transmit the load outward and will only accept the load from other failed nodes, then for any unfailed node $v_j$, there must be:

$$Q_j(0)\le Q_j(t-1)\le Q_j(t)$$

(10)

Then, for Formula (9), we have:

$$Q_j(0)+\frac{C_j}{{\displaystyle {\sum }_{\phi \in \Omega }{C}_{\phi }}}{Q}_i\le Q_j(t-1)+\frac{C_j}{{\displaystyle {\sum }_{\phi \in \Omega }{C}_{\phi }}}{Q}_i\le \beta S_j^{\alpha }$$

(11)

Further obtained:

$$S_j^{\alpha }+\frac{S_j^{\alpha }}{{\displaystyle {\sum }_{\phi \in \Omega }{S}_{\phi }^{\alpha }}}{S}_i^{\alpha }\le \beta S_j^{\alpha }$$

(12)

According to the derivation method of the cascading failure critical condition of the scale-free network in the literature [20], the relationship between the traffic intensity and the node degree of the taxi transportation network in this paper can be regarded as $S_i\sim d^{\theta }$, then the Formula (12) can be transformed into:

$$k_j^{\alpha \theta }+\frac{k_j^{\alpha \theta }}{{\displaystyle {\sum }_{\phi \in \Omega }{k}_{\phi }^{\alpha \theta }}}{k}_i^{\alpha \theta }\le \beta k_j^{\alpha \theta }$$

(13)

Due to the node reserve capacity coefficient $\beta \ge 1$, $\beta$ can be split into $\beta =(1+\gamma )$, $\gamma \ge 0$, and substituted into (13) to get:

$$\frac{k_i^{\alpha \theta }}{{\displaystyle {\sum }_{\phi \in \Omega }{k}_{\phi }^{\alpha \theta }}}\le \gamma$$

(14)

Because in complex network theory, degree and degree distribution satisfy Equations (15)–(17):

$${\displaystyle \sum _{k=k_{\min }}^{k_{\max }}P(k)}=1$$

(15)

$${\displaystyle \sum _{k=k_{\min }}^{k_{\max }}kP(k)}=\langle k\rangle$$

(16)

$${\displaystyle {\sum }_{\phi \in \Omega }{k}^{\alpha \theta }}={\displaystyle \sum _{k^{\prime }=k_{\min }}^{k_{\max }}{k}_{i}P(\frac{k^{\prime }}{k_i})}{k^{\prime }}^{\alpha \theta }$$

(17)

For the taxi transportation network, each OD information is independent, i.e., each traveler’s origin and destination are based on his own transportation needs and are not influenced by other travelers, which ensures that the network is degree uncorrelated, and therefore, Equation (18) also holds in the taxi transportation network:

$$P(\frac{k^{\prime }}{k_i})=\frac{k^{\prime }P(k^{\prime })}{\langle k\rangle }$$

(18)

where $P(\raisebox{1ex}{$k^{\prime }$}\!\left/ \!\raisebox{-1ex}{$k_i$}\right.)$ is the conditional probability that a node with degree value $k_i$ has an adjacent node with degree value $k^{\prime }$. Combining with the Formulas (14)–(18), we can finally deduce:

$$\frac{k_i^{\alpha \theta -1}\langle k\rangle }{\langle {k^{\prime }}^{\alpha \theta +1}\rangle }\le \gamma$$

(19)

Substitute the node traffic intensity into Formula (20).

$$\frac{S_i^{\frac{\alpha \theta -1}{\theta }}\langle S\rangle }{\langle S_{k^{\prime }}{}^{\frac{\alpha \theta +1}{\theta }}\rangle }\le \gamma$$

(20)

For a given network, $\gamma$, $\langle S\rangle$, and $\langle S_{k^{\prime }}{}^{(\alpha \theta +1)/\theta }\rangle$ are all fixed values. When $(\alpha \theta -1)/\theta \ge 0$, $S_i^{(\alpha \theta -1)/\theta }$ is a monotonically increasing function of $S_i$. When $S_i$ is larger, the inequality of Formula (20) does not hold. The actual performance is that when the high-load initial node fails, it is more likely to cause cascading failures. When $(\alpha \theta -1)/\theta <0$, $S_i^{(\alpha \theta -1)/\theta }$ is a monotonically decreasing function of $S_i$. When $S_i$ is smaller, the inequality of Formula (20) does not hold, that is, when the low-load initial node fails, it is more likely to cause cascading failures. When $\alpha =1/\theta$, the nodes to resist cascading failures is more well-distributed.

5. Simulation Experiments of Taxi Transportation Network

5.1. Simulation Parameter Setting

According to the above analysis, the influence of nodes failure is different under different parameters. We set up three groups of control simulation experiment scenarios, $\alpha <1/\theta$, $\alpha =1/\theta$, and $\alpha >1/\theta$. Each scenario considers different capacity coefficients $\gamma$. We attack all nodes of the taxi transportation network by the flow intensity, and record the $\gamma$ value of each node, which do not to have cascading failure.

Combining with the analysis results ($\theta \approx 1.37$) in Section 2.3, we can obtain $\alpha \approx 0.73$ when $\alpha =1/\theta$. Therefore, in the three groups of control experiments, the initial load distribution coefficients are set as $\alpha =0.20$, $\alpha =0.73$, and $\alpha =1.20$. Start at 0 with a step size of 0.005 and loop 100 times. Each set of experiments is simulated 10 times.

5.2. Simulation Results and Analysis

As shown in Figure 9, when $\alpha =0.20<1/\theta$, compared with high-load nodes, the node capacity required for low-load nodes to avoid cascading failures when attacked is larger, and the $\gamma$ value for individual nodes not to fail is as high as 0.4 or even 0.5. This means that the network is at $\alpha <1/\theta$, nodes with low initial load are vulnerable and more prone to cascading failures, while nodes with high initial load have better stability and a strong ability to resist cascading failures. The correlation between flow intensity and cascading failure does not seem to be reflected in Figure 10. When $\alpha =0.73=1/\theta$, the $\gamma$ value of the network without cascading failure is between 0.04 and 0.12, and the network cascading failure has no significant correlation with the initial load of the failed node. This means that the network is at $\alpha =1/\theta$ and the nodes to resist cascading failures are more uniform. Figure 11 is the exact opposite of Figure 9, when $\alpha =1.20>1/\theta$, nodes with lower initial load in the network are more stable, and a few nodes no longer have cascading failures whose $\gamma$ is 0.02. However, nodes with a higher initial load show a certain vulnerability which a higher $\gamma$ value is required to ensure that cascading failures will not occur.

Through further analysis, it can be found that when $\alpha <1/\theta$ or $\alpha >1/\theta$, there is a certain power-law distribution relationship between the cascading failure resistance of network nodes and the size of the initial load.

Considering the real scene, the high initial load ($\alpha >1/\theta$) represents the peak of urban taxi passenger flow, and the low initial load ($\alpha <1/\theta$) represents the flat peak or trough of urban taxi passenger flow. High-load nodes represent high-traffic-active areas in urban hotspots, and low-load nodes represent remote, low-traffic-active areas. According to the simulation results, the anti-interference ability of Lanzhou taxi network is quite good. However, during the peak period, if the passenger flow increases in urban hotspot areas, the taxi transportation networks will be prone to cascade failures, which will affect the taxi passenger service capacity in other areas of the city, and there will be a certain risk of network paralysis. During the peak or trough period, the taxi service capacity is sufficient in urban hotspot areas, and the capacity coordination and deployment capacity between hotspot areas is strong; whereas, the service capacity in remote urban areas is insufficient, and the capacities that can be coordinated and deployed between adjacent areas are relatively limited. Faced with a sudden surge in traffic demand, paying more attention to the taxi passenger service in remote urban areas is necessary.

6. Conclusions

In this paper, we extract the pick-up and drop-off point information of the taxi GPS trajectory data, use the clustering method to construct the taxi transportation network, and design a taxi passenger cascading failure model with the nonlinear function of the traffic intensity as the initial load. Then, the critical conditions of cascading failure are analyzed. The results show that the network construction method based on the clustering of pick-up and drop-off points can effectively reflect the topological characteristics of the taxi transportation network, which can be used as a powerful tool for mining the spatial and temporal distribution characteristics of taxi passenger flow. The cascading failure model and simulation process of the taxi transportation networks proposed in this paper also reveal the law of differences of taxi passenger service capacity under different passenger flow backgrounds.

Increasing the node capacity is an effective way to enhance the strength of the network and improve the anti-attack capability of the network. Still, the increase in the node capacity also means a rise in construction and operation costs. Therefore, moderately controlling node flow, and allocating network resources scientifically and rationally, are the most powerful way to avoid cascading failures in taxi transportation networks and improve taxi service capabilities.

For taxi transportation networks, the adjustment of nodes’ capacity and load is more abstract than the cascading failure problem of traditional bus or rail networks. In this paper, on the one hand, the node is only set in two states, normal and failed, and in future research, we are prepared to set more refined states, such as normal, saturated but not failed, and failed, to enrich our research. On the other hand, for the capacity of the node, it can be explained in several aspects such as regional vehicle density and taxi empty cruising rate, etc. If more data on taxi operations can be combined to make more accurate quantitative analysis of these elements in the future, this will bring more useful information to passengers and taxi operating companies.