# The Structure and First-Passage Properties of Generalized Weighted Koch Networks

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## Abstract

**:**

## 1. Introduction

## 2. The Generalized Weighted Koch Network

## 3. Topological Properties and ATT

#### 3.1. Degree Distribution

**Theorem**

**1.**

**Proof.**

#### 3.2. Clustering Coefficient

**Theorem**

**2.**

**Proof.**

#### 3.3. Diameter

**Theorem**

**3.**

**Proof.**

#### 3.4. Average Weighted Shortest Path

**Theorem**

**4.**

**Proof.**

#### 3.5. ATT on Random Walk with Weight

**Theorem**

**5.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Values of A1–A6

## Appendix B. The Values of B1–B6

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t | ${\mathit{k}}_{\mathit{i}}^{\mathit{A}}$ | ${\mathit{n}}^{\mathit{A}}\left({\mathit{k}}_{\mathit{i}}^{\mathit{A}}\right)$ | ${\mathit{k}}_{\mathit{i}}^{\mathit{B}}$ | ${\mathit{n}}^{\mathit{B}}\left({\mathit{k}}_{\mathit{i}}^{\mathit{B}}\right)$ |
---|---|---|---|---|

0 | 2 | $s(s-1){(s+1)}^{t-1}$ | $s-1$ | $s(s-1){(s+1)}^{t-1}$ |

1 | $2\times 2$ | $s(s-1){(s+1)}^{t-2}$ | $2(s-1)$ | $s(s-1){(s+1)}^{t-2}$ |

⋯ | ⋯ | ⋯ | ⋯ | ⋯ |

${t}_{i}$ | $2\times {2}^{{t}_{i}}$ | $s(s-1){(s+1)}^{t-{t}_{i}-1}$ | ${2}^{{t}_{i}}(s-1)$ | $s(s-1){(s+1)}^{t-{t}_{i}-1}$ |

⋯ | ⋯ | ⋯ | ⋯ | ⋯ |

$t-1$ | $2\times {2}^{t-1}$ | $s(s-1){(s+1)}^{0}$ | ${2}^{t-1}(s-1)$ | $s(s-1){(s+1)}^{0}$ |

t | $2\times {2}^{t}$ | s | ${2}^{t}(s-1)$ | s |

t | ${\mathit{k}}_{\mathit{i}}^{\mathit{B}}$ | $\mathit{c}\left({\mathit{k}}_{\mathit{i}}^{\mathit{B}}\right)$ | ${\mathit{n}}^{\mathit{B}}\left({\mathit{k}}_{\mathit{i}}^{\mathit{B}}\right)$ |
---|---|---|---|

0 | $s-1$ | 1 | $s(s-1){(s+1)}^{t-1}$ |

1 | $2(s-1)$ | $\frac{s-2}{2(s-1)-1}$ | $s(s-1){(s+1)}^{t-2}$ |

⋯ | ⋯ | ⋯ | ⋯ |

${t}_{i}$ | ${2}^{{t}_{i}}(s-1)$ | $\frac{s-2}{{2}^{{t}_{i}}(s-1)-1}$ | $s(s-1){(s+1)}^{t-{t}_{i}-1}$ |

⋯ | ⋯ | ⋯ | ⋯ |

$t-1$ | ${2}^{t-1}(s-1)$ | $\frac{s-2}{{2}^{t-1}(s-1)-1}$ | $s(s-1){(s+1)}^{1}$ |

t | ${2}^{t}(s-1)$ | $\frac{s-2}{{2}^{t}(s-1)-1}$ | s |

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**MDPI and ACS Style**

Su, J.; Zhang, M.; Yao, B.
The Structure and First-Passage Properties of Generalized Weighted Koch Networks. *Entropy* **2022**, *24*, 409.
https://doi.org/10.3390/e24030409

**AMA Style**

Su J, Zhang M, Yao B.
The Structure and First-Passage Properties of Generalized Weighted Koch Networks. *Entropy*. 2022; 24(3):409.
https://doi.org/10.3390/e24030409

**Chicago/Turabian Style**

Su, Jing, Mingjun Zhang, and Bing Yao.
2022. "The Structure and First-Passage Properties of Generalized Weighted Koch Networks" *Entropy* 24, no. 3: 409.
https://doi.org/10.3390/e24030409