# Research on the Node Importance of a Weighted Network Based on the K-Order Propagation Number Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Weighted K-Order Propagation Number Algorithm

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypotheses 1 and 3**, we can find the number of infected nodes ${N}_{{v}_{i}}^{K}$ after the propagation time of K, when setting ${v}_{i}$ as the source of infection:

## 3. Node Importance Analysis for the WKPN Algorithm Based on a Deliberate Attack Strategy

#### 3.1. A Symmetric Network with Bridge Nodes

#### 3.2. Real Networks

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Erdős, P.; Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci.
**1960**, 5, 17–60. [Google Scholar] - Wang, X.F.; Li, X.; Chen, G.R. Complex Network Theory and Its Application; Qing Hua University Publication: Beijing, China, 2006; p. 8. [Google Scholar]
- Lambiotte, R.; Rosvall, M.; Scholtes, I. From networks to optimal higher-order models of complex systems. Nat. Phys.
**2019**, 15, 313–320. [Google Scholar] [CrossRef] - Zhou, X.; Zhang, F.M.; Zhou, W.P.; Zou, W.; Yang, F. Evaluating complex network functional robustness by node efficiency. Acta Phys. Sin.
**2012**, 61, 190201. [Google Scholar] [CrossRef] - Zhou, T.; Bai, W.; Wang, B.; Liu, Z.J.; Yan, G. A brief review of complex networks. Physics
**2005**, 34, 31–36. [Google Scholar] [CrossRef] - Liu, J.G.; Wang, Z.T.; Dang, Y.Z. Optimization of robustness of scale-free network to random and targeted attacks. Mod. Phys. Lett. B
**2005**, 19, 785–792. [Google Scholar] [CrossRef] [Green Version] - Bharali, A.; Baruah, D. On network criticality in robustness analysis of a network structure. Malaya J. Mat. (MJM)
**2019**, 7, 223–229. [Google Scholar] [CrossRef] [Green Version] - Kunegis, J. Handbook of Network Analysis [KONECT–the Koblenz Network Collection]. arXiv
**2014**, arXiv:1402.5500, 1343–1350. [Google Scholar] - Lu, X.B.; Wang, X.F.; Li, X.; Fang, J.Q. Synchronization in weighted complex networks: Heterogeneity and synchronizability. Phys. A Stat. Mech. Its Appl.
**2006**, 370, 381–389. [Google Scholar] [CrossRef] - Zhou, Y.M.; Wang, J.W.; Huang, G.Q. Efficiency and robustness of weighted air transport networks. Transp. Res. Part E Logist. Transp. Rev.
**2019**, 122, 14–26. [Google Scholar] [CrossRef] - Hu, P.; Fan, W.; Mei, S. Identifying node importance in complex networks. Phys. A Stat. Mech. Its Appl.
**2015**, 429, 169–176. [Google Scholar] [CrossRef] - Freeman, L.C. Centrality in social networks conceptual clarification. Soc. Netw.
**1978**, 1, 215–239. [Google Scholar] [CrossRef] [Green Version] - Opsahl, T.; Agneessens, F.; Skvoretz, J. Node centrality in weighted networks: Generalizing degree and shortest paths. Soc. Netw.
**2010**, 32, 245–251. [Google Scholar] [CrossRef] - Gao, C.; Lan, X.; Zhang, X.G.; Deng, Y. A bio-inspired methodology of identifying influential nodes in complex networks. PLoS ONE
**2013**, 8, e66732. [Google Scholar] [CrossRef] - Garas, A.; Schweitzer, F.; Havlin, S. A k-shell decomposition method for weighted networks. New J. Phys.
**2012**, 14, 083030. [Google Scholar] [CrossRef] - Wang, B.; Ma, R.N.; Wang, G.; Chen, B. Improved evaluation method for node importance based on mutual information in weighted networks. Comput. Appl.
**2015**, 35, 1820. [Google Scholar] [CrossRef] - Zhao, S.X.; Rousseau, R.; Fred, Y.Y. h-Degree as a basic measure in weighted networks. J. Inf.
**2011**, 5, 668–677. [Google Scholar] [CrossRef] - Korn, A.; Schubert, A.; Telcs, A. Lobby index in networks. Phys. A Stat. Mech. Its Appl.
**2009**, 388, 2221–2226. [Google Scholar] [CrossRef] - Page, L.; Brin, S.; Motwani, R.; Winograd, T. The PageRank Citation Ranking: Bringing Order to the Web; Technical Report; Stanford InfoLab: Stanford, CA, USA, 1999. [Google Scholar]
- Zhao, L.; Xiong, L.; Xue, S. Global Recursive Based Node Importance Evaluation. In Advanced Data Mining and Applications; Springer International Publishing: Berlin/Heidelberg, Germany, 2016; pp. 738–750. [Google Scholar] [CrossRef]
- Wang, H.; Hernandez, J.M.; Van Mieghem, P. Betweenness centrality in a weighted network. Phys. Rev. E
**2008**, 77, 046105. [Google Scholar] [CrossRef] [Green Version] - Freeman, L.C. A set of measures of centrality based on betweenness. Sociometry
**1977**, 40, 35–41. [Google Scholar] [CrossRef] - Tizghadam, A.; Leon-Garcia, A. Betweenness centrality and resistance distance in communication networks. IEEE Netw.
**2010**, 24, 10–16. [Google Scholar] [CrossRef] - Pagani, G.A.; Aiello, M. The power grid as a complex network: A survey. Phys. A Stat. Mech. Its Appl.
**2013**, 392, 2688–2700. [Google Scholar] [CrossRef] [Green Version] - Maslov, S.; Sneppen, K.; Zaliznyak, A. Detection of topological patterns in complex networks: Correlation profile of the internet. Phys. A Stat. Mech. Its Appl.
**2004**, 333, 529–540. [Google Scholar] [CrossRef] - Lü, L.; Chen, D.; Ren, X.L.; Zhang, Q.M.; Zhang, Y.C.; Zhou, T. Vital nodes identification in complex networks. Phys. Rep.
**2016**, 650, 1–63. [Google Scholar] [CrossRef] [Green Version] - Kitsak, M.; Gallos, L.K.; Havlin, S.; Liljeros, F.; Muchnik, L.; Stanley, H.E.; Makse, H.A. Identification of influential spreaders in complex networks. Nat. Phys.
**2010**, 6, 888–893. [Google Scholar] [CrossRef] [Green Version] - Huang, L.Y.; Huo, Y.L.; Wang, Q.; Cheng, X.F. Network heterogeneity based on K-order structure entropy. Acta Phys. Sin.
**2019**, 68, 18901. [Google Scholar] [CrossRef] - Isella, L.; Stehlé, J.; Barrat, A.; Cattuto, C.; Pinton, J.; Van den Broeck, W. What’s in a Crowd? Analysis of Face-to-Face Behavioral Networks. J. Theor. Biol.
**2011**, 271, 166–180. [Google Scholar] [CrossRef] [Green Version] - Opsahl, T. Triadic closure in two-mode networks: Redefining the global and local clustering coefficients. Soc. Netw.
**2013**, 35, 159–167. [Google Scholar] [CrossRef] [Green Version] - Opsahl, T. Why Anchorage Is Not (That) Important: Binary Ties and Sample Selection. 2011. Available online: http://toreopsahl.com/2011/08/12/why-anchorage-is-not-that-important-binary-tiesand-sample-selection (accessed on 16 September 2019).
- Colizza, V.; Pastor-Satorras, R.; Vespignani, A. Reaction–diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys.
**2007**, 3, 276. [Google Scholar] [CrossRef] [Green Version] - Li, P.; Ren, Y.; Xi, Y. An importance measure of actors (set) within a network. Syst. Eng.
**2004**, 22, 13–20. [Google Scholar] [CrossRef] - He, N.; Li, D.Y.; Gan, W.Y.; Zhu, X. Mining vital nodes in complex networks. Comput. Sci.
**2007**, 34, 1–5. [Google Scholar] [CrossRef] - Zhao, Z.Y.; Meng, X.R.; Sun, R.N. Nodes Importance Ranking Method Based on Multi-attribute Evaluation and Deletion. Comput. Eng.
**2018**, 44, 62. [Google Scholar] [CrossRef] - Chen, D.; Lü, L.; Shang, M.S.; Zhang, Y.C.; Zhou, T. Identifying influential nodes in complex networks. Phys. A Stat. Mech. Its Appl.
**2012**, 391, 1777–1787. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**The networks graph structure: (

**a**) the Science Museum visitor network; (

**b**) the Facebook forum network; (

**c**) the non-US airport route network; and (

**d**) the US 500 busiest commercial airports network.

**Figure 3.**K-order structure entropy ${H}^{K}$ varies with K: (

**a**) the Science Museum visitor network; (

**b**) the Facebook forum network; (

**c**) the non-US airport route network; and (

**d**) the US 500 busiest commercial airports network.

**Figure 4.**The network efficiency decline rate varies with the attack times: (

**a**) the Science Museum visitor network; (

**b**) the Facebook forum network; (

**c**) the non-US airport route network; and (

**d**) the US 500 busiest commercial airports network.

**Figure 5.**The node number of maximum sub-graph $\gamma $ in the network varies with the number of attacks: (

**a**) the Science Museum visitor network; (

**b**) the Facebook forum network; (

**c**) the non-US airport route network; and (

**d**) the US 500 busiest commercial airports network.

**Table 1.**The node importance sorting results of the network, as shown in Figure 1.

Node No. | Weighted K-Order Propagation Number (WKPN) Algorithm | Mutual Information (MI) Algorithm | ||
---|---|---|---|---|

Node Importance | Sort | Node Importance | Sort | |

1 | 3.60 | 5 | $-1.96$ | 5 |

2 | 2.32 | 7 | $-2.77$ | 7 |

3 | 6.18 | 3 | 5.30 | 1 |

4 | 6.78 | 1 | 2.19 | 3 |

5 | 0 | 9 | $-2.77$ | 7 |

6 | 0 | 9 | $-2.77$ | 7 |

7 | 6.78 | 1 | 2.19 | 3 |

8 | 6.18 | 3 | 5.30 | 1 |

9 | 3.60 | 5 | $-1.96$ | 5 |

10 | 2.32 | 7 | $-2.77$ | 7 |

**Table 2.**Average efficiency of the network in Figure 1 before and after the corresponding node is deleted.

Network Characteristic | Initial Network | Deleting the Most Important Node, ${\mathit{v}}_{3}$ or ${\mathit{v}}_{8}$, in the MI Algorithm | Deleting the Most Important Node, ${\mathit{v}}_{4}$ or ${\mathit{v}}_{7}$ in the WKPN Algorithm |
---|---|---|---|

Average Efficiency e | 0.2931 | 0.2529 | 0.2084 |

Decline of Average Efficiency e | 0 | 0.0402 | 0.0847 |

Decline Rate $\epsilon $ | 0 | 13.72% | 28.90% |

**Table 3.**Basic features of the Science Museum visitor network, Facebook forum network, the non-US airport routing network, and the US 500 busiest commercial airports network, including the number of nodes N, the number of edges E, and a short description.

Name of the Network | N | E | Description |
---|---|---|---|

Science Museum visitor | 206 | 714 | Weight stating the number of face-to-face contacts between visitors in the Science Museum. |

Facebook forum | 899 | 71,380 | Nodes representing the forum users and the information communication between users and the weights of the edges indicating the number of pieces of information that have ever been sent. |

Non-US airport routing | 7976 | 15,250 | Demonstrating the routing structure between two non-US airports. |

US 500 busiest commercial airports | 500 | 2980 | Describing the structure of passengers traveling between the 500 busiest commercial airports. |

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## Share and Cite

**MDPI and ACS Style**

Tang, P.; Song, C.; Ding, W.; Ma, J.; Dong, J.; Huang, L.
Research on the Node Importance of a Weighted Network Based on the *K*-Order Propagation Number Algorithm. *Entropy* **2020**, *22*, 364.
https://doi.org/10.3390/e22030364

**AMA Style**

Tang P, Song C, Ding W, Ma J, Dong J, Huang L.
Research on the Node Importance of a Weighted Network Based on the *K*-Order Propagation Number Algorithm. *Entropy*. 2020; 22(3):364.
https://doi.org/10.3390/e22030364

**Chicago/Turabian Style**

Tang, Pingchuan, Chuancheng Song, Weiwei Ding, Junkai Ma, Jun Dong, and Liya Huang.
2020. "Research on the Node Importance of a Weighted Network Based on the *K*-Order Propagation Number Algorithm" *Entropy* 22, no. 3: 364.
https://doi.org/10.3390/e22030364