# Modeling and Complex Characteristics of Urban Subway Co-Opetition Network: A Case Study of Wuhan

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Research Status

#### 2.1. Co-Opetition Relationship of Urban Traffic

#### 2.2. Research on Complex Characteristics

## 3. Co-Opetition Relationship Analysis of Conventional Public Transport Networks

#### 3.1. Type Division of Co-Opetition Relationship

#### 3.2. Measurement of Co-Opetition Intensity

#### 3.2.1. Competition Index

#### 3.2.2. Cooperation Index

#### 3.2.3. Co-Opetition Index

## 4. Modeling of Urban Subway Co-Opetition Network

#### 4.1. Conventional Subway Co-Opetition Network Based on Space R

^{com}= <V,E

^{com}> and G

^{coo}= <V,E

^{coo}>, where G

^{com}is the competition network, G

^{coo}is the cooperation network, V = {L

_{1}、L

_{2}…L

_{N}} is the line set, N is the number of lines, E

^{com}= {${F}_{z/com}^{1-2}$、${F}_{z/com}^{1-3}$…${F}_{z/com}^{\mathrm{N}-1-\mathrm{N}}$} is the edge weight of the competition network, and E

^{coo}= {${F}_{z/coo}^{1-2}$、${F}_{z/coo}^{1-3}$…${F}_{z/coo}^{\mathrm{N}-1-\mathrm{N}}$} is the edge weight of the cooperation network. The urban subway co-opetition network is a directed weighted network.

#### 4.2. Complexity Metrics of the Co-Opetition Network

## 5. Case Study

#### 5.1. Characteristic Analysis of Competitive Network and Cooperative Network

^{2}of the current network is about 0.98 and 0.88. The goodness of fit R

^{2}of the long-term network is about 0.94 and 0.97. This result shows that the competition network and cooperation network of the Wuhan subway are scale-free networks. In Table 3, the diameter of the co-opetition network of the current network is shown to be two, and the clustering coefficients are 0.807 and 0.846. The diameter of the co-opetition network of the long-term network is two, and the clustering coefficients are 0.777 and 0.863, indicating high clustering with short-range small-world network characteristics.

#### 5.2. Characteristic Analysis of the Co-Opetition Network

^{2}of the current network co-opetition is about 0.88, and the goodness of fit R

^{2}of the long-term network competition is about 0.97. This shows that the co-opetition network of the Wuhan subway is a scale-free network. The diameter of the co-opetition network is two, and the clustering coefficients are 0.846 and 0.863, indicating high clustering and short-range small-world network characteristics.

## 6. Conclusions and Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Space R network. (

**a**) The network structure based on Space R; (

**b**) The network structure of Space R when considering the walkable transfer between D and E.

**Figure 8.**Cumulative distribution of cooperation and competition index intensity. (

**a**) Node competition index intensity distribution; (

**b**) node cooperation index intensity distribution.

**Figure 9.**Index intensity matrix of cooperation and competition between subway lines. (

**a-1**) Inter-line competition index intensity matrix of the current network; (

**a-2**) inter-line cooperation index intensity matrix of the current network; (

**b-1**) inter-line competition index intensity matrix of the long-term network; (

**b-2**) inter-line co-opetition index intensity matrix of the long-term network.

**Figure 12.**Index intensity matrix of co-opetition between subway lines. (

**a**) Inter-line co-opetition index intensity matrix of the current network; (

**b**) inter-line co-opetition index intensity matrix of the long-term network.

**Figure 13.**Scatter chart of competition and cooperation indexes between lines. (

**a**) Scatter chart of competition and cooperation indexes between current network lines; (

**b**) scatter chart of competition and cooperation indexes between long-term network lines.

Indexes | Formula |
---|---|

Node cooperation index intensity | ${k}_{i}{}^{coo}={\displaystyle \sum _{{I}^{l}}{F}_{z/coo}^{i-j}}$ |

Node competition index intensity | ${k}_{i}{}^{com}={\displaystyle \sum _{{I}^{l}}{F}_{z/com}^{i-j}}$ |

Node co-opetition index intensity | ${k}_{i}{}^{cc}={\displaystyle \sum _{{I}^{l}}{F}_{z/cc}^{i-j}}$ |

Node clustering coefficient | ${C}_{i}={E}_{i}/{C}^{2}{}_{{k}_{i}}$ |

Global clustering coefficient | $C={\displaystyle \sum _{i=1}^{N}{C}_{i}/N}$ |

Average path length | $D={\displaystyle \sum _{i\ne j}{d}_{ij}/N(N-1)}$ |

Network diameter | $L=\mathrm{max}{d}_{ij}$ |

Parameter | Explanation |
---|---|

${a}_{ij}$ | Adjacency matrix variables. When ${a}_{ij}=0$, i and j are not connected and i and j cannot be transferred. When ${a}_{ij}=1$, i and j are connected, i and j can transfer |

${I}^{i}$ | Set of nodes connected to node i |

${d}_{ij}$ | The shortest length between nodes i and j |

${E}_{i}$ | Actual number of connected edges of node i |

Network Type | Average Weighted Degree | Global Aggregation Coefficient C | Average Path Length D | Network Diameter L | |
---|---|---|---|---|---|

Current subway network | Competition network | 0.177 | 0.807 | 1.286 | 2 |

Cooperation network | 3.803 | 0.846 | 1.25 | 2 | |

Co-opetition network | 3.814 | 0.846 | 1.25 | 2 | |

Long-term subway network | Competition network | 0.29 | 0.777 | 1.473 | 2 |

Cooperation network | 7.577 | 0.863 | 1.242 | 2 | |

Co-opetition network | 7.597 | 0.863 | 1.242 | 2 |

Line Name | Current Network | Long-term Network | ||||||
---|---|---|---|---|---|---|---|---|

Cooperative Intensity | Ranking | Competitive Intensity | Ranking | Cooperative Intensity | Ranking | Competitive Intensity | Ranking | |

Line1 | 8.198 | 4 | 0.519 | 2 | 16.984 | 6 | 1.05 | 3 |

Line2 | 8.901 | 2 | 0.357 | 5 | 14.944 | 9 | 1.112 | 2 |

Line3 | 10.211 | 1 | 0.408 | 4 | 18.103 | 4 | 0.711 | 6 |

Line4 | 7.812 | 6 | 0.184 | 7 | 13.585 | 11 | 1.272 | 1 |

Line5 | - | - | - | - | 18.704 | 3 | 0.301 | 9 |

Line6 | 6.176 | 7 | 0.599 | 1 | 16.323 | 7 | 0.83 | 5 |

Line7 | 8.578 | 3 | 0.464 | 3 | 14.81 | 10 | 0.573 | 7 |

Line8 | 7.903 | 5 | 0.209 | 6 | 17.439 | 5 | 0.194 | 11 |

Line10 | - | - | - | - | 15.503 | 8 | 0.515 | 8 |

Line11 | - | - | - | - | 18.814 | 1 | 0.247 | 10 |

Line12 | - | - | - | - | 18.74 | 2 | 1.002 | 4 |

Line16 | - | - | - | - | 7.761 | 14 | 0.011 | 14 |

Line19 | - | - | - | - | 11.737 | 12 | 0.167 | 12 |

Line21 | 3.069 | 8 | 0.09 | 8 | 8.712 | 13 | 0.142 | 13 |

Line Name | Current Network | Long-Term Network | ||
---|---|---|---|---|

Co-Opetition | Ranking | Co-Opetition | Ranking | |

Line1 | 8.243 | 4 | 17.083 | 6 |

Line2 | 8.917 | 2 | 15.059 | 9 |

Line3 | 10.223 | 1 | 18.139 | 4 |

Line4 | 7.817 | 6 | 13.691 | 11 |

Line5 | - | - | 18.711 | 3 |

Line6 | 6.232 | 7 | 16.385 | 7 |

Line7 | 8.605 | 3 | 14.831 | 10 |

Line8 | 7.91 | 5 | 17.442 | 5 |

Line10 | - | - | 15.527 | 8 |

Line11 | - | - | 18.82 | 1 |

Line12 | - | - | 18.813 | 2 |

Line16 | - | - | 7.761 | 14 |

Line19 | - | - | 11.741 | 12 |

Line21 | 3.071 | 8 | 8.719 | 13 |

Classification | Division Basis | Quantity (Proportion) | ||
---|---|---|---|---|

Current Network | Long-Term Network | |||

Competitive type only | ${F}_{z/com}^{i-j}>0,{F}_{z/coo}^{i-j}=0$ | 0 (0%) | 0 (0%) | |

Conventional type | Competitive dominant type | ${F}_{z/com}^{i-j}>{F}_{z/coo}^{i-j}>0$ | 0 (0%) | 0 (0%) |

Cooperative dominant type | ${F}_{z/coo}^{i-j}>{F}_{z/com}^{i-j}>0$ | 20 (55.56%) | 54 (59.34%) | |

Cooperative type only | ${F}_{z/coo}^{i-j}>0,{F}_{z/com}^{i-j}=0$ | 1 (2.78%) | 15 (16.48%) | |

Unrelated type | ${F}_{z/com}^{i-j}=0,{F}_{z/coo}^{i-j}=0$ | 15 (41.67%) | 22 (24.18%) |

Line Name | Current Network | Long-Term Network | Deflection Angle Variation (°) | |||||
---|---|---|---|---|---|---|---|---|

Cooperation Deflection Angle (°) | Competition Deflection Angle (°) | Co-Opetition Effect | Cooperation Deflection Angle (°) | Competition Deflection Angle (°) | Co-Opetition Effect | Cooperative Deflection Angle Increment | Competition Deflection Angle Increment | |

Line 1 | 86.378 | 3.622 | 8.215 | 86.462 | 3.538 | 17.016 | 0.085 | −0.085 |

Line 2 | 87.7 | 2.3 | 8.909 | 85.745 | 4.255 | 14.985 | −1.956 | 1.956 |

Line 3 | 87.71 | 2.29 | 10.219 | 87.75 | 2.25 | 18.117 | 0.04 | −0.04 |

Line 4 | 88.648 | 1.352 | 7.815 | 84.651 | 5.349 | 13.644 | −3.997 | 3.997 |

Line 5 | 89.077 | 0.923 | 18.707 | 89.077 | 0.923 | |||

Line 6 | 84.457 | 5.543 | 6.205 | 87.088 | 2.912 | 16.344 | 2.63 | −2.63 |

Line 7 | 86.907 | 3.093 | 8.59 | 87.786 | 2.214 | 14.821 | 0.879 | −0.879 |

Line 8 | 88.487 | 1.513 | 7.906 | 89.361 | 0.639 | 17.44 | 0.874 | −0.874 |

Line 10 | 88.099 | 1.901 | 15.511 | 88.099 | 1.901 | |||

Line 11 | 89.247 | 0.753 | 18.816 | 89.247 | 0.753 | |||

Line 12 | 86.939 | 3.061 | 18.767 | 86.939 | 3.061 | |||

Line 16 | 89.921 | 0.079 | 7.761 | 89.921 | 0.079 | |||

Line 19 | 89.184 | 0.816 | 11.738 | 89.184 | 0.816 | |||

Line 21 | 88.312 | 1.688 | 3.07 | 89.067 | 0.933 | 8.713 | 0.755 | −0.755 |

Mean value | 87.325 | 2.675 | 7.616 | 87.884 | 2.116 | 15.17 | - |

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## Share and Cite

**MDPI and ACS Style**

Pan, Y.; Chang, M.; Feng, S.; Hao, D.
Modeling and Complex Characteristics of Urban Subway Co-Opetition Network: A Case Study of Wuhan. *Sustainability* **2023**, *15*, 883.
https://doi.org/10.3390/su15010883

**AMA Style**

Pan Y, Chang M, Feng S, Hao D.
Modeling and Complex Characteristics of Urban Subway Co-Opetition Network: A Case Study of Wuhan. *Sustainability*. 2023; 15(1):883.
https://doi.org/10.3390/su15010883

**Chicago/Turabian Style**

Pan, Yilei, Mengying Chang, Shumin Feng, and Dongsheng Hao.
2023. "Modeling and Complex Characteristics of Urban Subway Co-Opetition Network: A Case Study of Wuhan" *Sustainability* 15, no. 1: 883.
https://doi.org/10.3390/su15010883