# Influential Nodes Identification in Complex Networks via Information Entropy

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Spreading Model

#### 2.2. EnRenew Algorithm

Algorithm 1: EnRenew |

#### 2.3. Computational Complexity Analysis

#### 2.4. Performance Metrics

## 3. Results and Discussions

#### 3.1. An Example Network

#### 3.2. Data Description

#### 3.3. Analysis of Influence Range When Renewing

#### 3.4. Comparation with Benchmark Algorithms

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**It shows how the red node’s (node 1) entropy is calculated in detail. Node 1 has four neighbors from node 2 to node 5. Node 1’s information entropy is then calculated by ${E}_{1}={H}_{21}+{H}_{31}+{H}_{41}+{H}_{51}=0.32+0.37+0.27+0.35=1.31$.

**Figure 2.**This network consists of three communities at different scales. The first nine nodes selected by EnRenew are marked red. The network typically shows the rich club phenomenon, that is, nodes with large degree tend to be connected together.

**Figure 3.**The figure shows how EnRenew’s parameter l influences final affected scale $F\left({t}_{c}\right)$ on the six networks. Each subfigure shows experiment results on one network. p is the ratio of initial infected nodes. The results are obtained by averaging on 100 independent runs with spread rate $\lambda =1.5$ in SIR. With specific ratio of initial infected nodes p, larger final affected scale $F\left({t}_{c}\right)$ means more reasonable of the parameter l. The best parameter l differs from different networks. In real life application, l can be used as an tuning parameter.

**Figure 4.**This experiment compares different methods by final affected scale $F\left({t}_{c}\right)$ on the six networks. Each subfigure shows experiment results on one network. p is the ratio of initial infected nodes. The results are obtained by averaging on 100 independent runs with spread rate $\lambda =1.5$ in Susceptible-Infected-Recovered (SIR). With specific ratio of initial spreading nodes p, larger final affected scale $F\left({t}_{c}\right)$ indicates that the selected nodes are more advantageous to spreading. It can be seen that EnRenew surpasses all the other benchmark methods on the six networks. On the two small network, EnRenew nearly reaches the upper bound.

**Figure 5.**This experiment compares different methods regard to spreading speed. Each subfigure shows experiment results on one network. The ratio of initial infected nodes is 3% for CEnew, Email, Hamster and Router, 0.3% for Condmat and 0.03% for Amazon. The results are obtained by averaging on 100 independent runs with spread rate $\lambda =1.5$ in SIR. With the same spreading time t, larger $F\left(t\right)$ indicates larger influence scale in network, which reveals a faster spreading speed. It can be seen from the figures that EnRenew spreads apparently faster than other benchmark methods on all networks. On the small network CEnew and Email, EnRenew’s spreading speed is close to the upper bound.

**Figure 6.**This experiment tests algorithms’ effectiveness on different spreading conditions. Each subfigure shows experiment results on one network. The ratio of initial infected nodes is 3% for CEnew, Email, Hamster and Router, 0.3% for Condmat, and 0.03% for Amazon. The results are obtained by averaging on 100 independent runs. Different infected rate $\lambda $ of SIR can imitate different spreading conditions. EnRenew gets a larger final affected scale $F\left({t}_{c}\right)$ on different $\lambda $ than all the other benchmark methods, which indicates the proposed algorithm has more generalization ability to different spreading conditions.

**Figure 7.**This experiment analysis average shortest path length ${L}_{S}$ of nodes selected by different algorithms. Each subfigure shows experiment results on one network. p is the ratio of initial infected nodes. Generally speaking, larger ${L}_{S}$ indicates the selected nodes are more sparsely distributed in network. It can be seen that nodes selected by EnRenew have the apparent largest ${L}_{S}$ on five networks. It shows EnRenew tends to select nodes sparsely distributed.

**Table 1.**Experiment results on example network shown in Figure 2. The first three nodes selected by EnRenew are distributed in three communities. The best results of Community Distribution, Average Distance and $F\left({t}_{c}\right)$ are marked in bold.

Method | Initial Spreading Nodes | Community Distribution | Average Distance | $\mathit{F}\left({\mathit{t}}_{\mathit{c}}\right)$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | |||

Adaptive Degree [58] | 8 | 30 | 7 | 18 | 12 | 16 | 13 | 26 | 24 | 5 | 3 | 1 | 3.64 | 18.22 |

PageRank [42] | 7 | 8 | 30 | 18 | 12 | 13 | 26 | 29 | 22 | 4 | 3 | 2 | 4.33 | 17.56 |

h-index [29] | 18 | 12 | 13 | 17 | 9 | 8 | 30 | 16 | 15 | 6 | 1 | 1 | 2.75 | 15.64 |

k-shell [33] | 7 | 8 | 30 | 18 | 12 | 13 | 10 | 26 | 17 | 5 | 2 | 1 | 3.72 | 17.97 |

VoteRank [61] | 7 | 30 | 8 | 18 | 12 | 14 | 26 | 22 | 12 | 5 | 2 | 2 | 3.94 | 18.36 |

Greedy [58] | 17 | 6 | 27 | 21 | 16 | 32 | 1 | 11 | 23 | 4 | 2 | 2 | 4.92 | 21.33 |

EnRenew ($l=2$) | 8 | 30 | 18 | 7 | 16 | 22 | 14 | 28 | 29 | 4 | 3 | 2 | 4.27 | 18.81 |

Networks | n | m | $\langle \mathit{k}\rangle $ | ${\mathit{k}}_{\mathit{max}}$ | $\langle \mathit{c}\rangle $ |
---|---|---|---|---|---|

CEnew | 453 | 2025 | 8.94 | 237 | 0.646 |

1133 | 5451 | 9.62 | 71 | 0.22 | |

Hamster | 2426 | 16631 | 13.711 | 273 | 0.538 |

Router | 5022 | 6258 | 2.492 | 106 | 0.012 |

Condmat | 23133 | 93497 | 8.083 | 281 | 0.633 |

Amazon | 334863 | 925872 | 5.530 | 549 | 0.397 |

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## Share and Cite

**MDPI and ACS Style**

Guo, C.; Yang, L.; Chen, X.; Chen, D.; Gao, H.; Ma, J.
Influential Nodes Identification in Complex Networks via Information Entropy. *Entropy* **2020**, *22*, 242.
https://doi.org/10.3390/e22020242

**AMA Style**

Guo C, Yang L, Chen X, Chen D, Gao H, Ma J.
Influential Nodes Identification in Complex Networks via Information Entropy. *Entropy*. 2020; 22(2):242.
https://doi.org/10.3390/e22020242

**Chicago/Turabian Style**

Guo, Chungu, Liangwei Yang, Xiao Chen, Duanbing Chen, Hui Gao, and Jing Ma.
2020. "Influential Nodes Identification in Complex Networks via Information Entropy" *Entropy* 22, no. 2: 242.
https://doi.org/10.3390/e22020242