# Coreness Variation Rule and Fast Updating Algorithm for Dynamic Networks

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## Abstract

**:**

## 1. Introduction

## 2. $\mathcal{H}$ Operator Method for Coreness Calculation

## 3. Coreness Variation in Dynamic Networks

#### 3.1. DE Experiments

#### 3.1.1. Classification of Nodes with the Same Coreness

#### 3.1.2. Coreness Variation Rule and CHDE Algorithm

**Step 1**: Judge the type of node $i$. If $i\in T1$, then find out the connected subgraph ${V}_{i}\left(h\right)$ by the searching algorithm (Figure 4). Otherwise if $i\in T2$, the coreness of any node will not vary, and the CHDE algorithm ends.

**Step 2**: Update the coreness of all nodes in the connected subgraph ${V}_{i}\left(h\right)$ by the $\mathcal{H}$ operator method [29].

#### 3.2. AE Experiments

#### 3.2.1. Classification of Nodes with the Same Coreness

#### 3.2.2. Coreness Variation Rule and CHAE Algorithm

**Step1**: Judge the type of node $i$. If $i\in T4$, then find out the connected subgraph ${V}_{i}\left(h\right)$ by the searching algorithm (Figure 4). Otherwise, if $i\in T3$, ${c}_{i}^{AE}={c}_{i}+1=h+1$, then the CHAE algorithm ends.

**Step 2**: Set the coreness of all nodes in the connected subgraph ${V}_{i}\left(h\right)$ as $h+1$, then update the coreness of all nodes in ${V}_{i}\left(h\right)$ by the $\mathcal{H}$ operator method [29].

## 4. Case Study

#### 4.1. Computational Efficiency of CHDE Algorithm

#### 4.2. Computational Efficiency of CHAE Algorithm

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Basic topological characteristics of five real networks. $\overline{k}$ is the average degree; $L$ is the average shortest path length; $C$ is the average clustering coefficient.

Network | $\mathit{N}$ | $\mathit{M}$ | $\overline{\mathit{k}}$ | $\mathit{L}$ | $\mathit{C}$ |
---|---|---|---|---|---|

1133 | 5450 | 9.622 | 3.606 | 0.220 | |

US Power Grid | 4941 | 6594 | 2.669 | 18.990 | 0.080 |

Petster-Hamster | 2426 | 16,631 | 13.711 | 3.588 | 0.538 |

ca-AstroPh | 18771 | 198,050 | 21.102 | 4.194 | 0.631 |

loc-brightkite | 58,228 | 214,078 | 7.035 | 3.789 | 0.172 |

${\mathit{c}}_{\mathit{i}}\mathbf{<}{\mathit{c}}_{\mathit{j}}$ | ${\mathit{c}}_{\mathit{i}}\mathbf{=}{\mathit{c}}_{\mathit{j}}$ | ||||
---|---|---|---|---|---|

DE | (1) $i\in T1$ | (2) $i\in T2$ | (3) $i\in T1$ and $j\in T1$ | (4) $i\in T1$ and $j\in T2$ | (5) $i\in T2$ and $j\in T2$ |

AE | (1) $i\in T3$ | (2) $i\in T4$ | (3) $i\in T3$ and $j\in T3$ | (4) $i\in T3$ and $j\in T4$ | (5) $i\in T4$ and $j\in T4$ |

**Table 3.**Relative computational efficiency (${R}_{\mathcal{H},DE}$) of CHDE Algorithm for five real networks.

Real Network | ${\mathit{R}}_{\mathcal{H}\mathbf{,}\mathit{D}\mathit{E}}$ | |||
---|---|---|---|---|

${\mathit{c}}_{\mathit{i}}\mathbf{<}{\mathit{c}}_{\mathit{j}}$ | ${\mathit{c}}_{\mathit{i}}\mathbf{=}{\mathit{c}}_{\mathit{j}}$ | |||

$\mathit{i}\mathbf{\in}\mathit{T}\mathbf{1}$ | $\mathit{i}\mathbf{\in}\mathit{T}\mathbf{1}$ $\mathit{j}\mathbf{\in}\mathit{T}\mathbf{1}$ | $\mathit{i}\mathbf{\in}\mathit{T}\mathbf{1}$ $\mathit{j}\mathbf{\in}\mathit{T}\mathbf{2}$ | ||

BEST | 426.98 | 301.58 | 281.19 | |

AVERAGE | 200.89 | 50.80 | 31.73 | |

US Power Grid | BEST | 474.56 | 245.87 | 379.19 |

AVERAGE | 285.67 | 20.30 | 60.02 | |

Petster-Hamster | BEST | 772.38 | 801.65 | 697.01 |

AVERAGE | 409.58 | 229.86 | 145.83 | |

ca-AstroPh | BEST | 1185.59 | 1139.01 | 1079.72 |

AVERAGE | 648.10 | 251.57 | 186.91 | |

loc-brightkite | BEST | 986.95 | 1125.96 | 1066.01 |

AVERAGE | 526.01 | 685.42 | 416.63 |

**Table 4.**Relative computational efficiency (${R}_{\mathcal{H},AE}$) of CHAE Algorithm for five real networks.

Real Network | ${\mathit{R}}_{\mathcal{H}\mathbf{,}\mathit{A}\mathit{E}}$ | |||
---|---|---|---|---|

${\mathit{c}}_{\mathit{i}}\mathbf{<}{\mathit{c}}_{\mathit{j}}$ | ${\mathit{c}}_{\mathit{i}}\mathbf{=}{\mathit{c}}_{\mathit{j}}$ | |||

$\mathit{i}\mathbf{\in}\mathit{T}\mathbf{4}$ | $\mathit{i}\mathbf{\in}\mathit{T}\mathbf{3}$ $\mathit{j}\mathbf{\in}\mathit{T}\mathbf{4}$ | $\mathit{i}\mathbf{\in}\mathit{T}\mathbf{4}$ $\mathit{j}\mathbf{\in}\mathit{T}\mathbf{4}$ | ||

BEST | 157.18 | 238.72 | 227.78 | |

AVERAGE | 9.28 | 74.15 | 54.71 | |

US Power Grid | BEST | 291.71 | 312.42 | 285.64 |

AVERAGE | 8.93 | 80.33 | 56.70 | |

Petster-Hamster | BEST | 874.83 | 547.72 | 512.52 |

AVERAGE | 104.43 | 158.35 | 114.82 | |

ca-AstroPh | BEST | 1018.30 | 1055.64 | 1059.94 |

AVERAGE | 170.80 | 233.28 | 229.48 | |

loc-brightkite | BEST | 821.53 | 955.76 | 857.46 |

AVERAGE | 271.96 | 332.20 | 304.58 |

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**MDPI and ACS Style**

Gao, L.; Gao, G.; Ma, D.; Xu, L.
Coreness Variation Rule and Fast Updating Algorithm for Dynamic Networks. *Symmetry* **2019**, *11*, 477.
https://doi.org/10.3390/sym11040477

**AMA Style**

Gao L, Gao G, Ma D, Xu L.
Coreness Variation Rule and Fast Updating Algorithm for Dynamic Networks. *Symmetry*. 2019; 11(4):477.
https://doi.org/10.3390/sym11040477

**Chicago/Turabian Style**

Gao, Liang, Ge Gao, Dandan Ma, and Lida Xu.
2019. "Coreness Variation Rule and Fast Updating Algorithm for Dynamic Networks" *Symmetry* 11, no. 4: 477.
https://doi.org/10.3390/sym11040477