# How to Identify the Most Powerful Node in Complex Networks? A Novel Entropy Centrality Approach

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminaries

## 4. Preliminaries of Information Entropy

## 5. Model Description

#### 5.1. Computing on Local Influence

**Definition**

**1.**

#### 5.2. Computing on Indirect Influence

Algorithm 1 Influence calculating algorithm |

Input: A connected network represented by graph $G\left(V,E\right)$ with $V$ vertices and $E$ edges. Output: The overall influence of node $i$. for $i=1$ to $V$ do identify the subgraph ${G}_{i}$ constructed by node $i$ and its one-hop neighbors; calculate the local influence of node $i$ on its one-hop neighbors using Equation (7); calculate the indirect influence of node $i$ using Equation (11); calculate the overall influence of node $i$ using Equation (12); end for |

#### 5.3. Example Explanation

## 6. Performance Evaluation

## 7. Conclusions and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**(

**a**) Degree distributions of Erdos-Renyi network; (

**b**) Degree distributions of Watts-Stroggatz network; (

**c**) Degree distributions of Barabasi-Albert network.

**Figure 6.**The frequency of nodes with the same centrality value using different measures in the karate club network.

**Figure 7.**The frequency of nodes with the same centrality value using different measures in USAir club network.

**Figure 8.**The frequency of nodes with the same centrality value using different measures in Collaboration network.

**Figure 9.**The frequency of nodes with the same centrality value using different measures in Email network.

**Figure 10.**The frequency of nodes with the same centrality value using different measures in Erdos-Renyi network.

**Figure 11.**(

**a**) The frequency of nodes with the same centrality value using different measures in Watts-Stroggatz network; (

**b**) The frequency of nodes with the same centrality value using EC, BC, CC and the proposed measures in Watts-Stroggatz network.

**Figure 12.**The frequency of nodes with the same centrality value using different measures in Barabasi-Albert network.

**Figure 13.**Comparing the spreading capacity of single node in karate club network between the proposed model and EC, PR.

**Figure 14.**(

**a**) A comparison of the spreading capacity of single node in USAir network between the proposed model and EC, PR; (

**b**) A comparison of the spreading capacity of single node in USAir network between the proposed model and BC.

**Figure 15.**(

**a**) A comparison of the spreading capacity of single node in Collaboration network between the proposed model and BC; (

**b**) A comparison of the spreading capacity of single node in Collaboration network between the proposed model and CC; (

**c**) A comparison of the spreading capacity of single node in Collaboration network between the proposed model and EC; (

**d**) A comparison of the spreading capacity of single node in Collaboration network between the proposed model and PR.

**Figure 16.**(

**a**) A comparison of the spreading capacity of single node in Email network between the proposed model and BC; (

**b**) A comparison of the spreading capacity of single node in Email network between the proposed model and CC; (

**c**) A comparison of the spreading capacity of single node in Email network between the proposed model and EC; (

**d**) A comparison of the spreading capacity of single node in Email network between the proposed model and PR.

**Figure 17.**(

**a**) A comparison of the spreading capacity of single node in Erdos-Renyi network between the proposed model and BC; (

**b**) A comparison of the spreading capacity of single node in Erdos-Renyi network between the proposed model and CC; (

**c**) A comparison of the spreading capacity of single node in Erdos-Renyi network between the proposed model and EC; (

**d**) A comparison of the spreading capacity of single node in Erdos-Renyi network between the proposed model and PR.

**Figure 18.**(

**a**) A comparison of the spreading capacity of single node in Watts-Stroggatz network between the proposed model and BC; (

**b**) A comparison of the spreading capacity of single node in Watts-Stroggatz network between the proposed model and CC; (

**c**) A comparison of the spreading capacity of single node in Watts-Stroggatz network between the proposed model and EC; (

**d**) A comparison of the spreading capacity of single node in Watts-Stroggatz network between the proposed model and PR.

**Figure 19.**(

**a**) A comparison of the spreading capacity of single node in Barabasi-Albert network between the proposed model and BC; (

**b**) A comparison of the spreading capacity of single node in Barabasi-Albert network between the proposed model and CC; (

**c**) A comparison of the spreading capacity of single node in Barabasi-Albert network between the proposed model and EC; (

**d**) A comparison of the spreading capacity of single node in Barabasi-Albert network between the proposed model and PR.

Node | $\mathit{S}\mathit{D}{\mathit{C}}_{\mathit{i}}$ |
---|---|

1 | 3 |

2 | 2 |

4 | 2 |

6 | 1 |

Node | Local Influence | Indirect Influence | Total Influence |
---|---|---|---|

1 | 0.5737 | 0.3584 | 0.4876 |

2 | 0.6836 | 0.4510 | 0.5906 |

3 | 0.5737 | 0.3934 | 0.5016 |

4 | 0.5933 | 0.4193 | 0.5237 |

5 | 0.7457 | 0.4439 | 0.6250 |

6 | 0.5737 | 0.3785 | 0.4956 |

7 | 0.5737 | 0.3820 | 0.4970 |

8 | 0.4515 | 0.2590 | 0.3745 |

Node | No. |
---|---|

5 | 1 |

2 | 2 |

4 | 3 |

3 | 4 |

7 | 5 |

6 | 6 |

1 | 7 |

8 | 8 |

**Table 4.**The top-10 ranked nodes of karate club network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Karate Club Network | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 34 | 1 | 1 | 34 | 34 | 1 |

2 | 1 | 34 | 3 | 1 | 1 | 34 |

3 | 33 | 33 | 34 | 3 | 33 | 33 |

4 | 3 | 3 | 32 | 33 | 3 | 3 |

5 | 2 | 32 | 9 | 2 | 2 | 2 |

6 | 4 | 9 | 14 | 9 | 32 | 4 |

7 | 32 | 2 | 33 | 14 | 4 | 32 |

8 | 9 | 14 | 20 | 4 | 24 | 9 |

9 | 14 | 20 | 2 | 32 | 9 | 14 |

10 | 24 | 6 | 4 | 31 | 14 | 24 |

**Table 5.**The top-10 ranked nodes of USAir97 network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

USAir97 Network | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 118 | 118 | 118 | 118 | 118 | 118 |

2 | 261 | 8 | 261 | 261 | 261 | 261 |

3 | 255 | 261 | 67 | 255 | 182 | 255 |

4 | 182 | 201 | 255 | 182 | 152 | 152 |

5 | 152 | 47 | 201 | 152 | 255 | 182 |

6 | 230 | 182 | 182 | 230 | 230 | 230 |

7 | 166 | 255 | 47 | 112 | 166 | 166 |

8 | 67 | 152 | 248 | 67 | 201 | 147 |

9 | 112 | 313 | 166 | 166 | 67 | 112 |

10 | 201 | 13 | 112 | 147 | 8 | 67 |

**Table 6.**The top-10 ranked nodes of the collaboration network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Collaboration Network | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 33 | 78 | 78 | 33 | 78 | 33 |

2 | 34 | 150 | 281 | 34 | 33 | 34 |

3 | 78 | 516 | 150 | 54 | 34 | 54 |

4 | 54 | 281 | 756 | 53 | 281 | 53 |

5 | 294 | 216 | 301 | 132 | 294 | 78 |

6 | 62 | 34 | 151 | 133 | 216 | 62 |

7 | 216 | 756 | 34 | 134 | 54 | 219 |

8 | 219 | 301 | 131 | 561 | 96 | 216 |

9 | 281 | 131 | 759 | 562 | 150 | 294 |

10 | 53 | 203 | 1123 | 840 | 46 | 281 |

**Table 7.**The top-10 ranked nodes of e-mail network URV by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

E-mail Network URV | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 105 | 333 | 333 | 105 | 105 | 105 |

2 | 333 | 105 | 23 | 16 | 23 | 16 |

3 | 16 | 23 | 105 | 196 | 333 | 42 |

4 | 23 | 578 | 42 | 204 | 41 | 196 |

5 | 42 | 76 | 41 | 42 | 42 | 3 |

6 | 41 | 233 | 76 | 49 | 16 | 333 |

7 | 196 | 135 | 233 | 56 | 233 | 49 |

8 | 233 | 41 | 52 | 116 | 355 | 41 |

9 | 21 | 355 | 135 | 333 | 21 | 354 |

10 | 76 | 42 | 378 | 3 | 24 | 332 |

**Table 8.**The top-10 ranked nodes of Erdos-Renyi network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Erdos-Renyi Network | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 74 | 74 | 74 | 74 | 74 | 74 |

2 | 88 | 88 | 63 | 88 | 88 | 88 |

3 | 23 | 23 | 23 | 63 | 23 | 23 |

4 | 99 | 99 | 88 | 39 | 99 | 63 |

5 | 68 | 68 | 68 | 68 | 68 | 68 |

6 | 25 | 63 | 39 | 23 | 46 | 39 |

7 | 46 | 39 | 26 | 69 | 39 | 46 |

8 | 63 | 42 | 99 | 25 | 25 | 5 |

9 | 39 | 76 | 27 | 26 | 63 | 6 |

10 | 42 | 13 | 2 | 27 | 86 | 99 |

**Table 9.**The top-10 ranked nodes of Watts-Stroggatz network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Watts-Stroggatz Network | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 466 | 466 | 466 | 466 | 466 | 183 |

2 | 183 | 198 | 54 | 183 | 183 | 466 |

3 | 221 | 54 | 291 | 185 | 162 | 398 |

4 | 54 | 130 | 130 | 428 | 221 | 428 |

5 | 322 | 13 | 128 | 429 | 322 | 221 |

6 | 162 | 162 | 198 | 54 | 198 | 54 |

7 | 198 | 221 | 167 | 198 | 54 | 97 |

8 | 404 | 11 | 433 | 76 | 271 | 172 |

9 | 11 | 183 | 211 | 467 | 294 | 198 |

10 | 13 | 128 | 13 | 369 | 63 | 454 |

**Table 10.**The top-10 ranked nodes of Barabasi-Albert network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Barabasi-Albert Network | ||||||
---|---|---|---|---|---|---|

Rank | DC | BC | CC | EC | PR | Proposed Model |

1 | 2 | 2 | 2 | 2 | 2 | 2 |

2 | 0 | 0 | 0 | 0 | 0 | 0 |

3 | 9 | 9 | 9 | 4 | 9 | 3 |

4 | 4 | 4 | 3 | 3 | 4 | 9 |

5 | 5 | 3 | 4 | 9 | 5 | 4 |

6 | 3 | 6 | 6 | 6 | 3 | 5 |

7 | 11 | 5 | 11 | 5 | 11 | 6 |

8 | 6 | 11 | 1 | 11 | 10 | 14 |

9 | 10 | 10 | 5 | 7 | 6 | 10 |

10 | 14 | 14 | 10 | 60 | 14 | 11 |

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**MDPI and ACS Style**

Qiao, T.; Shan, W.; Zhou, C.
How to Identify the Most Powerful Node in Complex Networks? A Novel Entropy Centrality Approach. *Entropy* **2017**, *19*, 614.
https://doi.org/10.3390/e19110614

**AMA Style**

Qiao T, Shan W, Zhou C.
How to Identify the Most Powerful Node in Complex Networks? A Novel Entropy Centrality Approach. *Entropy*. 2017; 19(11):614.
https://doi.org/10.3390/e19110614

**Chicago/Turabian Style**

Qiao, Tong, Wei Shan, and Chang Zhou.
2017. "How to Identify the Most Powerful Node in Complex Networks? A Novel Entropy Centrality Approach" *Entropy* 19, no. 11: 614.
https://doi.org/10.3390/e19110614