# Characterization of Symmetry of Complex Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Group Actions

- $f({g}_{1},f({g}_{2},s))=f({g}_{1}{g}_{2},s)$, for ${g}_{1},{g}_{2}\in G$ and $s\in S$,
- $f(e,s)=s$, where e denotes the identity element of the group G.

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

## 3. Symmetry Indexes

#### 3.1. Group Actions Arise from Networks

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 3.2. Symmetry Indexes for Networks

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Definition**

**1.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Example**

**2.**

## 4. Comparison to the Existing Symmetry Index

## 5. Application to Real-World Networks

## 6. Conclusions

## 7. Notations

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the symmetry indexes. Vertices with the same color belong to the same orbit.

Networks | (V, E) | $\mathbf{\left|}\mathbf{Aut}\mathbf{\right(}\mathsf{\Gamma}\mathbf{\left)}\mathbf{\right|}$ | ${\mathbf{SI}}_{\mathsf{\Gamma}\mathbf{,}\mathbf{1}}$ | ${\mathbf{SI}}_{\mathsf{\Gamma}\mathbf{,}\mathbf{2}}$ | ${\mathbf{SI}}_{\mathsf{\Gamma}\mathbf{,}\mathbf{3}}$ | ${\mathit{\beta}}_{\mathsf{\Gamma}}$ | ${\mathit{\gamma}}_{\mathsf{\Gamma}}$ |
---|---|---|---|---|---|---|---|

Karate Club | (34, 78) | 480 | 0.1471 | 0.2 | 0.2518 | 0.089 | 0.3235 |

Dolphins | (62, 159) | 4 | 0.0323 | 0.5 | 0.5167 | 0.0427 | 0.0645 |

Word Adjacency | (112, 425) | 2 | 0.01786 | 0.5 | 0.5 | 0.0237 | 0.0179 |

Football Game | (115, 613) | 1 | 0.0087 | 1 | 1 | 0.2297 | 0 |

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**MDPI and ACS Style**

Chen, Y.; Zhao, Y.; Han, X.
Characterization of Symmetry of Complex Networks. *Symmetry* **2019**, *11*, 692.
https://doi.org/10.3390/sym11050692

**AMA Style**

Chen Y, Zhao Y, Han X.
Characterization of Symmetry of Complex Networks. *Symmetry*. 2019; 11(5):692.
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**Chicago/Turabian Style**

Chen, Yangyang, Yi Zhao, and Xinyu Han.
2019. "Characterization of Symmetry of Complex Networks" *Symmetry* 11, no. 5: 692.
https://doi.org/10.3390/sym11050692