4.1. Threshold Analysis for Cascading Failure
We first investigate the thresholds to trigger maximum cascading failures under different attacks. Different levels of attack have been simulated by adding different
R to different stations. In
Figure 4, we plot the variation of balanced failure proportion I with the increasing
R in the four chosen Wednesdays of the experiment dataset. As shown in
Figure 4, when
, failures start to be triggered in the metro topology network. However, RA and IAs obviously have different trigger thresholds. As
R increases, the curve of CA will first make a mutation, and then that of XZ, SRS and RA. The results show that the large-scale cascading failure of IA is more easily triggered than RA because only a relatively small attack is needed to realize the rapid spread of cascading failures over the network. Thus, the Shanghai metro network is more robust to RFs than to IAs. This mutation effect also shows that for important stations, even a very small increase in the perturbation value
R can quickly trigger large-scale cascading failure of metro stations. The results reveal the inherent vulnerability of the metro as a special mean of transportation.
From
Figure 4, we observe that the stations near CA and SRS are densely distributed, while the stations near XZ are relatively dispersed. Therefore, CA and SRS have a larger flow pressure than XZ. From
Figure 4,c, before the balanced
I of XZ is held constant, there is a relatively large decrease as
R increases. This result indicates that when the value of
R reaches a certain threshold, the importance of CA and SRS to the whole metro network is higher than that of XZ. However, this phenomenon is not significant in
Figure 4a,d. We observe that the curves have a similar trend at the first and last Wednesday of April. However, there is a significant difference in the second and third Wednesday of April. The results show that mid-month heterogeneity of stations is more significant than that at the beginning and end of the month, and the importance level has a stratified effect. It reveals the significant temporal–spatial heterogeneity of cascading failures in metro systems.
Moreover, we find that in all the subgraphs of
Figure 4, the balanced
I of each station increases firstly and then decreases instead of increasing gradually to 1. The maximum value of balanced
I is close to 0.8, which demonstrates that after adding the topology coupling strength and passenger flow coupling strength, the Shanghai metro system is impossible to fail completely. Previous studies whose experiments guaranteed complete failure of the whole metro system as long as the perturbation
R is large enough generally assume that
[
19,
20,
21,
23]. Analyzing the mathematical formulas (e.g., Formulas (7) and (8)), the status value
of the next time step of a station is almost a fixed composition structure (when
): (1) 50% * the t step status value
; (2) 25% * the topology information of the neighbor node; and (3) 25% * the passenger flow information of the neighbor node.
This means that as long as the of the failed neighbor node is large enough, even multiplying by 0.25 will lead to the node failure. However, in Formula (10), if a station has a large number of neighbors (e.g., 7 neighbors), the and of the neighboring site can be very small (e.g., topology coupling strength = 1/7). This will lead to that the of the neighbor node in the next time step is mostly determined by the state of the node in the current time step . This result shows that in the metro system loaded with real passenger flow, when the failure characteristic information is transmitted to a metro station with many adjacent stations, the state of the next time step determined by the formula is mostly borne by the state of the current time step (e.g., 80% or 90%), and the other parts jointly constitute the rest (e.g., 20% or 10%). This explains why the global failure does not occur even under very large perturbation from one initial failed station. When one station has a large number of neighbor nodes, no matter how serious a neighbor node fails, all of the neighbors share part of its next time step status information .
By comparing
Figure 4 and
Figure 5, we find that the curve of the morning peak is different from that of the evening peak. For example, the similarity of the four curves in
Figure 4b,c is higher than that in
Figure 4a,d. However, the similarity of the four curves in
Figure 5a,d is higher than that in
Figure 5b,c. It implies that for CA and SRS, the balanced
I always increases fast when
, then takes a little drop, finally remains constant. For XZ, the mutation rules of the morning and evening peak curves vary from time to time in a month. Most mutations occur in the interval (1, 2) and the amplitudes of them are larger. For RAs, a fast increase of cascading failure relatively requires a larger
R than that of IAs.
The critical perturbation Rs at different time are shown in
Table 3. For example,
for XZ fluctuate mainly around 4,
for CA fluctuate mainly around 5,
for SRS fluctuate mainly around 6, and
s for RA fluctuate mainly around 7. The results indicate that the scale of cascading failure changes with
R, and it also changes with time, however, the perturbation in the station that leads to steady
is approximately a constant.
4.2. Cascading Failure Process
In
Figure 6, we plot the cumulative failure proportion
as a function of time step
t. When
R is small, for example,
, the failure will spread in a short time and disappear quickly, and the maximum
remains relatively small value, even less than 0.05. The result accords with the actual situation in which a low-level accident on a station will cause other limited metro stations to fail in a short time. It is also easier for the metro authorities to solve these problems. However, when there is a serious accident, the failure will gradually spread over the metro network. For example, in
Figure 5a, when
, the
of the curve with
increases to 0.04 while that of the curve with
increases to 0.3.
Compared with the curves of XZ shown in
Figure 6d, the curves of SRS and CA shown in
Figure 6b,c obviously trigger the maximum network failure from initial failure (
) more easily. SRS gets a steady value at
, CA gets that at
, while XZ gets that at
. Combined with the distribution of the three metro stations labeled in
Figure 1, we find that the cascading failure triggered from a station surrounded by other densely distributed stations needs fewer steps to spread over the metro network.
Compared with the curves of RA shown in
Figure 6a, the curves of IAs are more centralized when for larger
R. For instance, the curves with
in
Figure 6b, the curves with
in
Figure 6c and the curves with
in
Figure 6d. Moreover, when
is small, the curves are similar with each other, especially for those with
. The result shows that it is easier for cascading failures to be predicted in the beginning and be increasingly unmanageable over time because the destruction of the metro network will restrict the passenger flow evacuation.
It can be seen from
Figure 7 that there is an approximate normal distribution visually between instant cascading failure speed and time step
t, which is verified by the Kolmogorov–Smirnov normal distribution test in
Table 4. If
p value >0.05, the data is normally distributed. When
R is small, the peak times of curves under both RA and IAs are always within two steps. It indicates that the cascading failure triggered by a slight accident or attack will be limited in a small-world network and terminate quickly. However, the peak time will significantly be delayed as
R increases to a certain value. It implies that the propagation of cascading failures is mainly reflected in the early stage. For example,
in
Figure 7a,
in
Figure 7b,
in
Figure 7c, and
in
Figure 7d.
In
Figure 7c, the curves of
for different
R are almost similar with each other, which indicates that SRS has the predictability to perturbations, at least when
. For IAs, we can see that XZ has several peak times (
), which is different from SRS and CA shown in
Figure 7b,c. We also observe that the
of XZ’s peak time is almost half of that of CA and SRS. It indicates that the station location has a great influence on the cascade failure speed. For RA shown in
Figure 7a, the curves have a relatively wider peak time range. The results show that different attack can significantly influence the cascading failure process. There is a diversity for the cascading failure proportion distribution and peak time as
R changes. When attacking a random station on a metro system, the attack often needs to be more intense, which means that a larger
R should be added. This echoes the previous conclusion.