# Dynamic Evolution Analysis of Complex Topology and Node Importance in Shenzhen Metro Network from 2004 to 2021

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Complex Topology Modeling of Metro Network

#### 2.2. Statistical Measurement of Node Centrality

#### 2.3. Identification and Ranking of Node Importance

## 3. Numerical Analysis and Discussions

#### 3.1. Dynamic Evolution Analysis of Network Topology

#### 3.2. Dynamic Evolution Analysis of Node Centrality

**DC**. The $D{C}_{i}$ is proportional to the importance of the node (i) in the network. Figure 4a shows the frequency-distribution histogram of the node degree (${k}_{i}$) in the SZMN at different periods. Nodes with $k=2$ account for the highest proportion in each period, indicating that the scale-free characteristics of the SZMN were more obvious. That is, a few hub nodes played leading roles in the network operation. The proportion of nodes with $k=2$ changed with an N-type trend over time. Moreover, it reached the maximum of 82.2% in 2011, then dropped to 76.5% in 2016, and finally rose to 78.3% in 2021. In contrast, the proportion of nodes with $k=4$ exhibited the И type, with the lowest proportion of 2.04% in 2010, and the highest proportion of 13.3% in 2016. Meanwhile, nodes with $k=8$ (namely, four-line transfer stations) began to appear in 2016, and their proportion shows a decreasing trend year by year. In general, the SZMN shows the characteristics of a small-world network with a small number of high-degree nodes (hubs) [16,21,59]. With the development of the network, the SZMN tended to be developed, and the proportion of low-degree nodes gradually decreased. Moreover, after removing the nodes with $k=1$, the fitting result of the accumulative degree clearly shows the truncated power-law distribution ($p\left(k\right)~{k}^{-\lambda}$) in the log–log coordinate system. The fitting accuracy (${R}^{2}$) is 0.98 and above (as plotted in Figure 4b). The $\lambda $ decreased from 7.32 in 2010 to 3.67 in 2016, and then stayed around 3.8. This, combined with the fact that the proportion of nodes with low degrees ($k=1,2)$ decreased from 94% to 83%, shows that the scale-free [60] and heterogeneous [4] characteristics of the SZMN become more and more evident with the expansion of the network.

**EC**. For the node (i), the $E{C}_{i}$ measures the sum of the importance of its adjacent nodes [8]. The frequency-distribution histograms and normal fitting curves of the EC in different periods are plotted in Figure 5. It can be clearly seen that the normal fitting curves of the EC frequency distribution in 2004 and 2009 were flatter, with a larger mean and standard deviation. Based on the EC distribution in each period, we concluded that the EC in the initial stage of the network was generally large and then tended to decrease. However, some key stations, such as Chegongmiao, the Convention & Exhibition Center, Civic Center, Futian, etc., were still high in EC compared with the others at any period, which is owing to the fact that the adjacent nodes around these keys were also more important. Nodes with high ECs tended to make up the key areas of the network.

**BC**. According to Figure 2, the $B{C}_{i}$ is used to determine the middlemost node in the network, which plays a bridge role in the network. Nodes with larger BCs represent more critical nodes (such as hub nodes) in the network [4,17,46]. The heat maps of the BC distribution and the frequency-distribution histograms under each period of the SZMN are shown in Figure 6a,b. The nodes with high BCs in each period are multiline transfer stations, and their control over the physical network is stronger than ordinary stations. For example, Chegongmiao Station ($\mathrm{ID}=10$) became the most critical hub of the whole network after changing from an ordinary station to a four-line transfer station in 2016. Similarly, Futian Station ($\mathrm{ID}=49$) changed from a two-line transfer station to a three-line transfer station, which resulted in the rise in the status of the network. Based on the BC frequency-distribution histograms of different periods, the network-development period can be divided into two stages, including $\left[2004,2011\right]$ and $\left[2016,2021\right]$. As plotted in Figure 6b, the normal fitting curves are also divided into two groups according to the shape. Three new lines (L-11/7/9) opened in 2016, and they had greater influences on the entire network topology.

**CC**. The node (i) has high $C{C}_{i}$ if the shortest distance to any other node in the graph (G) is small [60]. The CC is closer to the geometric centrality than the BC. From Figure 7a, we can clearly see the heat map of the CC distribution of all the nodes in the SZMN. With the dynamic evolution, the nodes at the geometric center of the network changed from Exhibition Center ($\mathrm{ID}=7$) to Futian ($\mathrm{ID}=49$) and Children’s Palace ($\mathrm{ID}=63$). Moreover, the region formed around these three stations can be considered the geometric center range of the SZMN. However, the center position, in the geometric sense, is not exactly equal to the functional center of the network. It is defective to judge the node importance according to the single centrality index, which is made up in Section 3.3 by integrating the multiple indicators. On the whole, the CCs of all the nodes have almost the same development trend over time. Similar to the DC/EC/BC, the fitting curves of the CC frequency-distribution histogram of all nodes (as shown in Figure 7b) also show oblate and flat trends for 2004 and 2009. Subsequently, the number of nodes increased, and the shortest travel distance from one node to other nodes became shorter. Such a network development trend is reasonable for the time being.

**PR**. The PR is derived from the webpage rank and is often used to evaluate the webpage optimization, emphasizing that a more important webpage is often referenced by other webpages. In the traffic-complex network, the PR is used to identify the key nodes that have greater impacts on the traffic efficiency in the traffic system [30]. Because the PRs of the nodes in the early stage of the network differed greatly from those in the later stage, the PRs of all nodes in each period were normalized based on Equation (1) for better comparison. Thus, all the PRs of all nodes were between $\left[0,1\right]$, and the specific results are shown in Figure 8a. With the development of the network, the node with the highest PR was always the Convention & Exhibition Center Station ($\mathrm{ID}=7$) from 2004 to 2010, and it became Shenzhen North Station ($\mathrm{ID}=94$) in 2011. From 2016 to 2021, Chegongmiao Station ($\mathrm{ID}=10$) owns the highest PR due to its four-line transfer ability. It can be seen that the nodes with the highest PRs are all multiline transfer stations. At the same time, these transfer nodes with high DCs and BCs often play greater control roles in the network and affect other nodes around them. Therefore, these key nodes should be managed and controlled to prevent the vulnerability caused by the deliberate attack. The normal fitting curves of the PR frequency distribution shown in Figure 8b are similar to the four index curves mentioned above, which indicates that the network was very unstable in the early stage of construction.

#### 3.3. Dynamic Evolution Analysis of Node Importance

## 4. Conclusions and Future Work

- (1)
- With the spatiotemporal evolution of the network, the SZMN gradually developed from a loop network to a tree network after 2011, and the number of loops grew linearly. The nodes in the SZMN became more and more intensive. Moreover, the proportion of low-degree nodes declined gradually, and the small-world effect was increasingly weakened. For the information transmission between nodes, the global efficiency decreased over time, but the local efficiency became higher. The fault-tolerant ability of the SZMN became stronger and the network became more and more assortative;
- (2)
- The proportion of high-degree nodes gradually increased, and the scale-free and heterogeneous characteristics of the SZMN become more and more obvious. The nodes with high ECs tended to form the core areas of the network. The nodes with high BCs in each period are all multiline transfer stations, and their control over the physical network is stronger. The three new lines that opened in 2016 (L-11/7/9) had a significant impact on the network topology. The CCs of all the nodes had the same overall development trend over time. Generally, the DCs, BCs and PRs of the transfer stations in the network were usually at a higher level, which should be focused on management to prevent the vulnerability caused by deliberate attack. The shortest travel distance from one node to others became shorter with the network development, and the evolution trend tended to be reasonable;
- (3)
- In the node-importance evaluation, the multi-attribute decision-making method is better than a single attribute. The EC occupies the highest influence weight of the five indicators. With the evolution over time, the node importance of the SZMN gradually dispersed from the core area of Chegongmiao–Futian to the direction of the Airport and Shenzhen North (high-speed railway station). So far, the network development trend looks rational, and it can avoid the vulnerability caused by deliberate attack. Moreover, the node importance is closely related to the changes in the node type, surrounding nodes and network environment. Thus, we should consider the spatiotemporal development trend of the network and the changes in the importance of adjacent nodes when evaluating the metro node importance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The topology structure of SZMN from 2004 to 2021: (

**a**) 2004; (

**b**) 2009; (

**c**) 2010; (

**d**) 2011; (

**e**) 2016; (

**f**) 2019; (

**g**) 2020; (

**h**) 2021.

**Figure 4.**(

**a**) The frequency-distribution histogram of node degree (${k}_{i}$) in SZMN; (

**b**) the truncated power-law distributions of accumulative degree in SZMN.

**Figure 6.**(

**a**) The heat maps of BC distribution in SZMN; (

**b**) the frequency-distribution histograms of BC distribution in SZMN.

**Figure 7.**(

**a**) The heat maps of CC distribution in SZMN; (

**b**) the frequency-distribution histograms of CC distribution in SZMN.

**Figure 8.**(

**a**) The heat maps of PR distribution in SZMN; (

**b**) the frequency-distribution histograms of PR distribution in SZMN.

**Figure 9.**The heat map of node-importance-ranking results of SZMN from 2004 to 2021: (

**a**) 2004; (

**b**) 2009; (

**c**) 2010; (

**d**) 2011; (

**e**) 2016; (

**f**) 2019; (

**g**) 2020; (

**h**) 2021.

No. | Periods | Existing Lines |
---|---|---|

1 | 2004 | L-1/4 |

2 | 2009 | L-1/4 |

3 | 2010 | L-1/2/3/4 |

4 | 2011 | L-1/2/3/4/5 |

5 | 2016 | L-1/2/3/4/5/7/9/11 |

6 | 2019 | L-1/2/3/4/5/7/9/11 |

7 | 2020 | L-1/2/3/4/5/6/7/8/9/10/11 |

8 | 2021 | L-1/2/3/4/5/6/7/8/9/10/11/20 |

Index | Definition | Formula |
---|---|---|

$DC$ [28] | $DC$ measures the total number of connected edges of a node. | ${k}_{i}={\displaystyle \sum}_{j}^{N}{\alpha}_{ij}$$,\text{}D{C}_{i}=\frac{{k}_{i}}{N-1}$ |

$EC$ [8,28] | $EC$ can identify the different effects of neighbor ones on a node on it. | $\lambda {e}_{i}={\displaystyle {\displaystyle \sum}_{j=1}^{N}}{\alpha}_{ij}{e}_{j}$ $e={\left[{e}_{1},{e}_{2},\cdots {e}_{n}\right]}^{T}$ |

$BC$ [29] | $BC$ is the shortest number of paths through a node. | $B{C}_{i}={\displaystyle {\displaystyle \sum}_{i\ne j\ne k}}\frac{{\sigma}_{jk}\left(i\right)}{{\sigma}_{jk}}$ |

$CC$ [29] | $CC$ is used to measure the ability of a station to affect another node through the network. | $C{C}_{i}=\frac{N-1}{{{\displaystyle \sum}}_{j=1,i\ne j}^{N}{d}_{ij}}$ |

$PR$ [30] | $PR$ is used to calculate the ranking of nodes in a graph based on the structure of incoming links. | $PR\left(i\right)=\left(1-\lambda \right)\frac{1}{n}+\lambda {\displaystyle {\displaystyle \sum}_{j:j\to k}}\frac{PR\left(j\right)}{\overline{{d}_{j}}}$ |

Period | N | E | L | β | γ | APL | D | ρ | $\mathit{\delta}$ | $\mathit{\theta}$ | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

2004 | 18 | 17 | 2 | 0.944 | 1 | 5.19 | 14 | 0.1111 | 0 | 0.3179 | −0.2289 |

2009 | 22 | 21 | 2 | 0.955 | 1 | 6.16 | 17 | 0.0909 | 0 | 0.2799 | −0.0194 |

2010 | 49 | 47 | 4 | 0.959 | 2 | — | — | 0.0400 | 0 | 0.1281 | −0.0066 |

2011 | 118 | 126 | 5 | 1.068 | 13 | 13.62 | 43 | 0.0183 | 0.0042 | 0.1239 | 0.1534 |

2016 | 166 | 190 | 8 | 1.145 | 32 | 11.64 | 43 | 0.0139 | 0.0026 | 0.1323 | −0.0431 |

2019 | 181 | 207 | 8 | 1.144 | 34 | 11.70 | 43 | 0.0127 | 0.0024 | 0.1292 | −0.0283 |

2020 | 236 | 271 | 11 | 1.148 | 46 | 13.60 | 42 | 0.0098 | 0.0037 | 0.1110 | 0.0603 |

2021 | 240 | 275 | 12 | 1.146 | 47 | 13.70 | 42 | 0.0096 | 0.0036 | 0.1099 | 0.0608 |

Period | $\overline{\mathit{D}\mathit{C}}$ | $\overline{\mathit{E}\mathit{C}}$ | $\overline{\mathit{B}\mathit{C}}$ | $\overline{\mathit{C}\mathit{C}}$ | $\overline{\mathit{P}\mathit{R}}$ |
---|---|---|---|---|---|

2004 | 0.1111 | 0.1763 | 0.2618 | 0.2036 | 0.0556 |

2009 | 0.0909 | 0.1475 | 0.2580 | 0.1715 | 0.0455 |

2010 | 0.0400 | — | 0.0854 | 0.0719 | 0.0204 |

2011 | 0.0183 | 0.0328 | 0.1088 | 0.0771 | 0.0085 |

2016 | 0.0139 | 0.0315 | 0.0649 | 0.0915 | 0.0060 |

2019 | 0.0127 | 0.0294 | 0.0598 | 0.0908 | 0.0055 |

2020 | 0.0098 | 0.0248 | 0.0538 | 0.0783 | 0.0042 |

2021 | 0.0096 | 0.0244 | 0.0534 | 0.0776 | 0.0042 |

${\mathit{W}}_{\mathit{j}}$ | DC | EC | BC | CC | PR |
---|---|---|---|---|---|

2004 | 0.139 | 0.353 | 0.303 | 0.09 | 0.116 |

2009 | 0.121 | 0.404 | 0.29 | 0.087 | 0.098 |

2010 | 0.155 | — | 0.569 | 0.152 | 0.125 |

2011 | 0.077 | 0.65 | 0.166 | 0.049 | 0.057 |

2016 | 0.094 | 0.527 | 0.257 | 0.052 | 0.07 |

2019 | 0.09 | 0.521 | 0.274 | 0.049 | 0.067 |

2020 | 0.088 | 0.553 | 0.245 | 0.053 | 0.062 |

2021 | 0.087 | 0.551 | 0.249 | 0.052 | 0.061 |

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**MDPI and ACS Style**

Meng, Y.; Qi, Q.; Liu, J.; Zhou, W.
Dynamic Evolution Analysis of Complex Topology and Node Importance in Shenzhen Metro Network from 2004 to 2021. *Sustainability* **2022**, *14*, 7234.
https://doi.org/10.3390/su14127234

**AMA Style**

Meng Y, Qi Q, Liu J, Zhou W.
Dynamic Evolution Analysis of Complex Topology and Node Importance in Shenzhen Metro Network from 2004 to 2021. *Sustainability*. 2022; 14(12):7234.
https://doi.org/10.3390/su14127234

**Chicago/Turabian Style**

Meng, Yangyang, Qingjie Qi, Jianzhong Liu, and Wei Zhou.
2022. "Dynamic Evolution Analysis of Complex Topology and Node Importance in Shenzhen Metro Network from 2004 to 2021" *Sustainability* 14, no. 12: 7234.
https://doi.org/10.3390/su14127234