# Network Vulnerability Analysis of Rail Transit Plans in Beijng-Tianjin-Hebei Region Considering Connectivity Reliability

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Network Model

#### 3.2. Critical Node Identification

#### 3.2.1. Node Importance Measure Based on Network Centrality

#### 3.2.2. Node Connectivity Reliability Measure Based on Monte Carlo Simulation

#### 3.2.3. Critical Nodes Identification Considering Connectivity Reliability

#### 3.3. Main Measures of Network Performance

- Efficiency $\mathrm{E}\left(k\right)$$$\mathrm{E}\left(k\right)=\frac{1}{\mathrm{N}\left(\mathrm{N}-1\right)}{\displaystyle \sum}_{i\ne j}\frac{1}{{\mathrm{d}}_{\mathrm{ij}}\left(k\right)}$$
- Origin–destination consider efficiency $\mathrm{ODE}\left(k\right)$

- 3.
- Largest component size LCS(k)$$\mathrm{LCS}\left(k\right)=\frac{{\mathrm{N}}_{\mathrm{l}}\left(k\right)}{\mathrm{N}}$$
- 4.
- Connectivity level CL(k)$$\mathrm{CL}\left(k\right)={\langle \frac{{\mathrm{N}}_{\mathrm{o}}^{\mathrm{i}}}{{\mathrm{N}}_{\mathrm{o}}}\rangle}_{\mathrm{i}}$$
- 5.
- Network average clustering coefficient CC(k)$$\mathrm{CC}\left(k\right)=\frac{1}{\mathrm{N}}{{\displaystyle \sum}}^{\text{}}{\mathrm{C}}_{\mathrm{i}}\left(k\right)$$

#### 3.4. Attacks Simulation

#### 3.5. Vulnerability Assessment Model

## 4. Application and Result

#### 4.1. Identifying the Critical Nodes

_{i}& B

_{i}and with higher connectivity probability still remain high rankings. Suppose the link connectivity probability p

_{ij}is known, the core-nodes ranking can be obtained, of which 15 nodes are listed here according to the limited space of paper. Therefore, the core-nodes ranking can be used as the attack sequence destroying the simulation when evaluating the network vulnerability.

#### 4.2. Simulations and Result Analysis

#### 4.2.1. Rail Transit Network within Region Analysis

_{c−n}, A

_{b}and A

_{bcr}. After 6 nodes (12% of nodes) are destroyed, E(k) decreases to less than 50%; (b) The decreasing patterns of E(k) are relatively similar under attack modes of A

_{b}and A

_{bcr}, or A

_{d}and A

_{dcr}. This is due to the close relationship of A

_{bcr}and A

_{dcr}with A

_{b}and A

_{d}, respectively. Significant difference would appear only when differences of node connectivity reliability is large; (c) Under 5 attack modes other than random failure, the change of network performance is relatively similar after two nodes, Beijing and Tianjin, are destroyed. The finding is consistent with common sense as Beijing and Tianjin are extremely critical points in the network. After the two cities are destroyed, the betweenness node-based attacks have larger impact on network efficiency, which are, however, beginning to change slower after the 8th node is destroyed. Meanwhile, the network efficiency decreases quickly under the degree node-based attacks. Such findings indicate that the nodes with a more direct link to other nodes have a higher sensitivity to the network efficiency than the nodes with more number of shortest paths passing through, after the former 8 nodes are destroyed.

#### 4.2.2. Regional Exterior Transit Analysis

_{r}and A

_{c−n}are carried out based on the measure of $\mathrm{ODE}\left(k\right)$ in order to evaluate the network vulnerability. Simulation outputs are generated in Figure 6, which shows that the network efficiency of links between three inland nodes and the estuary changes a lot under the attack mode of A

_{c−n}. After the first two critical nodes are destroyed, the network efficiency decreases to zero considering OD (origin-destination). The finding indicates that transit links between three inland cities and Yujiabao will all be destroyed after Beijing and Tianjin are unable to provide service. Especially once Tianjin station is destroyed, the transit network will be invalid as a whole. Robustness remains in the network under random failure.

## 5. Conclusions

- (1)
- The critical node identification considering node connectivity reliability is based on connectivity probability of network links. As the output nodes from Monte Carlo is based on the measure of network centrality, the sequencing of critical nodes is similar with that of network centrality when the centrality and connectivity reliability are relatively high. However, when the connectivity reliability is low, the method of critical node identification can take network centrality into consideration as well as the real connectivity reliability.
- (2)
- The attack modes based on random failure and 5 strategic modes provide simulations of different forms of destructions on the BTHR rail transit plan network. The network performances of E(k) and LCS(k) are simulated, and the results indicate that the network retains robustness under random failure. Under strategic attacks, though, the network shows ability to resist attack, the network vulnerability is relatively higher. An interesting finding from the measure of ODE(k) concerning regional exterior transit demand shows that failure of two critical nodes (Beijing and Tianjin) would cause fatal effect on the whole network.
- (3)
- The critical node rankings are quite different under different measures, such as degree/betweenness node-based metrics that have various emphases. However, when evaluating the network vulnerability, different aspects of influence should be taken into consideration. The network performance simulation under attack mode of Core-Nodes provides relatively balanced outputs between measures based on degree and on betweenness, from the perspective of either E(k) or LCS(k). Therefore, the measure of Core-Nodes is more suitable for critical node identification, as it represents for comprehensive network performance.
- (4)
- Although both centralities of Beijing and Tianjin are high, the influence of their failures on the whole BTHR rail transit network is only 6–21%. However, with multiple nodes failures, especially when the 8th node is destroyed, have huge impact on network performance, and the impact on E(k) is larger than that of LCS(k). The finding indicates that the robustness in the BTHR rail transit plan network is quite strong, though the impact of critical nodes failure on shortest paths is relatively high, the influence on partial nodes connectivity is quite small.
- (5)
- Considering regional network performance, the protection and emergency rescue preparation are not only essential for several large nodes such as Beijing and Tianjin, but is also important for nodes of Hengshui, Shijiazhuang, Miyun, Cangzhou, Huailai and Tangshan which have huge impact on the whole network shortest paths and connectivity. In terms of the regional exterior transit or transit towards the sea, Tianjin becomes a life-and-death node which should be paid large attention to. The safety protection and emergency rescue preparation should be strengthened, and multiple branch links connected to the sea should be constructed in order to raise the robustness of the network.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Description of the degree of BTHR rail system development plans nodes; (

**b**) Description of the betweenness of the BTHR rail system development plans nodes.

**Table 1.**Station and node number of Beijing-Tianjin-Hebei Region (BTHR) rail system development plans network.

Node No. | Station | Node No. | Station | Node No. | Station | Node No. | Station |
---|---|---|---|---|---|---|---|

1 | Beijing | 14 | Qinhuangdao | 27 | Zhangxin | 40 | Huangye |

2 | Langfang | 15 | Chongli | 28 | Tongzhou | 41 | Hejian |

3 | Tianjin | 16 | Xiahuayuan | 29 | Yizhuang | 42 | Dingzhou |

4 | Yujiapu | 17 | Zhangjiakou | 30 | Huangcun | 43 | Anxin |

5 | Sea-front | 18 | Huailai | 31 | Liangxiang | 44 | Baodi |

6 | Shijiazhuang | 19 | Miyun | 32 | Chengde | 45 | Jixian |

7 | Xingtai | 20 | Pinggu | 33 | Shenyang direction | 46 | Zunhua |

8 | Handan | 21 | Xianghe | 34 | Huhehaote direction | 47 | Qian’an |

9 | Baigou | 22 | Wuqing North | 35 | Taiyuan direction | 48 | Laoting |

10 | Bazhou | 23 | New airport | 36 | Hengshui | 49 | Caofeidian |

11 | Baoding | 24 | Gu’an | 37 | Ji’nan direction | ||

12 | Cangzhou | 25 | Zhuozhou | 38 | Zhengzhou direction | ||

13 | Tangshan | 26 | Capital airport | 39 | Liaocheng direction |

Network | N | E | <k> | C | B |
---|---|---|---|---|---|

Value | 49 | 83 | 3.388 | 0.169 | 67.571 |

Symbol | Description |
---|---|

d_{ij} (k) | The shortest path between node i and node j |

${\mathrm{N}}_{\mathrm{o}}$ | Set of origin nodes of the planning rail network, representing, origin stations |

${\mathrm{N}}_{\mathrm{d}}$ | Set of destination nodes of the planning rail network, representing, destination stations |

${\mathrm{N}}_{\mathrm{l}}\left(k\right)$ | The number of the nodes in the largest connected sub- network |

${\mathrm{N}}_{\mathrm{o}}^{\mathrm{i}}$ | The number of nodes in the fraction connected with origin node |

${\mathrm{C}}_{\mathrm{i}}\left(k\right)$ | A node of degree at least 2 as the proportion of links between the vertices within its neighborhood divided by the number of links that could possibly exist between the neighbors [36] |

$\mathrm{E}\left(k\right)$ | The normalized average value of the inverse of shortest path distance tween any two nodes |

$\mathrm{ODE}\left(k\right)$ | Only considers the shortest path between the origin nodes and the destination nodes |

LCS(k) | The ratio of nodes to total nodes in the largest connected sub-network to total nodes |

CL(k) | The average fraction of nodes of origin nodes connected by each node |

$\mathrm{CC}\left(k\right)$ | The average clustering coefficient measures the clustering (triangulation) within a network by averaging the clustering coefficients of all its nodes. |

Symbol | Description |
---|---|

${A}_{r}$ | Attack the nodes on random order |

${A}_{d}$ | Attack the nodes on the order of ${\mathrm{K}}_{\mathrm{i}}$ |

${A}_{b}$ | Attack the nodes on the order of ${\mathrm{B}}_{\mathrm{i}}$ |

${A}_{dcr}$ | Attack the nodes on the order of ${\mathrm{DCR}}_{\mathrm{i}}$ |

${A}_{bcr}$ | Attack the nodes on the order of ${\mathrm{BCR}}_{\mathrm{i}}$ |

${A}_{c-n}$ | Attack the nodes on the order of Core-nodes |

Node | Degree | Node | Betweenness |
---|---|---|---|

Beijing | 9 | Tianjin | 309.269 |

Tianjin | 7 | Beijing | 306.662 |

Hengshui | 6 | Huailai | 191.767 |

Langfang | 5 | Shijiazhuang | 159.752 |

Miyun | 5 | Miyun | 155.514 |

Yizhuang | 5 | Hengshui | 143.881 |

Tangshan | 5 | Cangzhou | 139.691 |

Qinhuangdao | 5 | Xiahuayuan | 137.000 |

Shijiazhuang | 5 | Liangxiang | 128.385 |

Cangzhou | 5 | Sea-front | 121.778 |

Tongzhou | 4 | Baodi | 111.573 |

Baoding | 4 | Bazhou | 93.558 |

Bazhou | 4 | Tangshan | 92.127 |

Baodi | 4 | Xingtai | 92.000 |

Baigou | 4 | Langfang | 89.827 |

Node | ${\mathsf{\omega}}_{\mathbf{i}}$ | Node | ${\mathsf{\omega}}_{\mathbf{i}}$ | Node | ${\mathsf{\omega}}_{\mathbf{i}}$ | Node | ${\mathsf{\omega}}_{\mathbf{i}}$ |
---|---|---|---|---|---|---|---|

1 | 0.931959 | 14 | 0.953820 | 27 | 0.95382 | 40 | 0.94389 |

2 | 0.929461 | 15 | 0.728314 | 28 | 0.95382 | 41 | 0.94389 |

3 | 0.953820 | 16 | 0.73031 | 29 | 0.95382 | 42 | 0.94389 |

4 | 0.914424 | 17 | 0.728314 | 30 | 0.95382 | 43 | 0.93195 |

5 | 0.953820 | 18 | 0.953820 | 31 | 0.95382 | 44 | 0.95382 |

6 | 0.943898 | 19 | 0.953820 | 32 | 0.95382 | 45 | 0.95382 |

7 | 0.943898 | 20 | 0.95382 | 33 | 0.95382 | 46 | 0.95382 |

8 | 0.908131 | 21 | 0.95382 | 34 | 0.723514 | 47 | 0.95382 |

9 | 0.931959 | 22 | 0.906539 | 35 | 0.914424 | 48 | 0.95382 |

10 | 0.943898 | 23 | 0.95382 | 36 | 0.943898 | 49 | 0.95382 |

11 | 0.931959 | 24 | 0.931959 | 37 | 0.943898 | ||

12 | 0.943898 | 25 | 0.931959 | 38 | 0.908131 | ||

13 | 0.95382 | 26 | 0.95382 | 39 | 0.943898 |

**Table 7.**Values of ${\mathrm{DCR}}_{\mathrm{i}}$ ${\mathrm{BCR}}_{\mathrm{i}}$ and the Core-Node ranking.

Node | $\mathbf{D}\mathbf{C}{\mathbf{R}}_{\mathbf{i}}$ | Node | ${\mathbf{K}}_{\mathbf{i}}$ | Node | $\mathbf{B}\mathbf{C}{\mathbf{R}}_{\mathbf{i}}$ | Node | ${\mathbf{B}}_{\mathbf{i}}$ | Core-Node | |
---|---|---|---|---|---|---|---|---|---|

1 | 8.387633 | 1 | 9 | 3 | 294.9871 | 3 | 309.269 | 1 | Beijing |

3 | 6.676743 | 3 | 7 | 1 | 285.7965 | 1 | 306.662 | 3 | Tianjin |

36 | 5.663388 | 36 | 6 | 18 | 182.9113 | 18 | 191.767 | 36 | Hengshui |

13 | 4.769102 | 2 | 5 | 6 | 150.7896 | 6 | 159.752 | 6 | Shijiazhuang |

14 | 4.769102 | 6 | 5 | 19 | 148.3324 | 19 | 155.514 | 19 | Miyun |

19 | 4.769102 | 12 | 5 | 36 | 135.809 | 36 | 143.881 | 12 | Cangzhou |

29 | 4.769102 | 13 | 5 | 12 | 131.854 | 12 | 139.691 | 18 | Huailai |

6 | 4.71949 | 14 | 5 | 31 | 122.4562 | 16 | 137 | 13 | Tangshan |

12 | 4.71949 | 19 | 5 | 5 | 116.1543 | 31 | 128.385 | 5 | Sea-front |

2 | 4.647306 | 29 | 5 | 44 | 106.4206 | 5 | 121.778 | 2 | Langfang |

5 | 3.815282 | 5 | 4 | 16 | 100.0525 | 44 | 111.573 | 14 | Qinhuangdao |

18 | 3.815282 | 9 | 4 | 10 | 88.30921 | 10 | 93.558 | 29 | Yizhuang |

21 | 3.815282 | 10 | 4 | 13 | 87.87261 | 13 | 92.127 | 10 | Bazhou |

23 | 3.815282 | 11 | 4 | 7 | 86.83861 | 7 | 92 | 44 | Baodi |

27 | 3.815282 | 18 | 4 | 2 | 83.49071 | 2 | 89.827 | 31 | angxiang |

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## Share and Cite

**MDPI and ACS Style**

Liu, J.; Lu, H.; Ma, H.; Liu, W.
Network Vulnerability Analysis of Rail Transit Plans in Beijng-Tianjin-Hebei Region Considering Connectivity Reliability. *Sustainability* **2017**, *9*, 1479.
https://doi.org/10.3390/su9081479

**AMA Style**

Liu J, Lu H, Ma H, Liu W.
Network Vulnerability Analysis of Rail Transit Plans in Beijng-Tianjin-Hebei Region Considering Connectivity Reliability. *Sustainability*. 2017; 9(8):1479.
https://doi.org/10.3390/su9081479

**Chicago/Turabian Style**

Liu, Jing, Huapu Lu, He Ma, and Wenzhi Liu.
2017. "Network Vulnerability Analysis of Rail Transit Plans in Beijng-Tianjin-Hebei Region Considering Connectivity Reliability" *Sustainability* 9, no. 8: 1479.
https://doi.org/10.3390/su9081479