#
A Generalization of the Importance of Vertices for an Undirected Weighted Graph^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

#### Graphs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6**

## 3. Measuring Vertex Importance on an Undirected Weighted Graph

**Definition**

**7**

**Definition**

**8**

**Definition**

**9**

## 4. Comparison and Analysis Results

#### 4.1. Data and Source Code

#### 4.2. Evaluation

#### 4.3. Ranking DIL-W${}^{\alpha}$ for Different Values of $\alpha $

#### 4.4. Computational Complexity

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Newman, M. The structure and function of complex networks. SIAM Rev.
**2003**, 45, 167–256. [Google Scholar] [CrossRef][Green Version] - Almasi, S.; Hu, T. Measuring the importance of vertices in the weighted human disease network. PLoS ONE
**2019**, 14, e0205936. [Google Scholar] [CrossRef][Green Version] - An, X.L.; Zhang, L.; Li, Y.Z.; Zhang, J.G. Synchronization analysis of complex networks with multi-weights and its application in public traffic network. Phys. A Stat. Mech. Its Appl.
**2014**, 412, 149–156. [Google Scholar] [CrossRef] - Manríquez, R.; Guerrero-Nancuante, C.; Martínez, F.; Taramasco, C. Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19. Int. J. Environ. Res. Public Health
**2021**, 18, 4432. [Google Scholar] [CrossRef] - Crossley, N.; Mechelli, A.; Vértes, P.; Winton-Brown, T.; Patel, A.; Ginestet, C.; McGuire, P.; Bullmore, E. Cognitive relevance of the community structure of the human brain functional coactivation network. Proc. Natl. Acad. Sci. USA
**2013**, 110, 11583–11588. [Google Scholar] [CrossRef][Green Version] - Wang, G.; Cao, Y.; Bao, Z.Y.; Han, Z.X. A novel local-world evolving network model for power grid. Acta Phys. Sin.
**2009**, 6, 58. [Google Scholar] - Montenegro, E.; Cabrera, E.; González, J.; Manríquez, R. Linear representation of a graph. Bol. Soc. Parana. Matemática
**2019**, 37, 97–102. [Google Scholar] [CrossRef][Green Version] - Pastor-Satorras, R.; Vespignani, A. Epidemic Spreading in Scale-Free Networks. Phys. Rev. Lett.
**2001**, 86, 3200–3203. [Google Scholar] [CrossRef][Green Version] - 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] - Vinterbo, S.A. Privacy: A machine learning view. IEEE Trans. Knowl. Data Eng.
**2004**, 16, 939–948. [Google Scholar] [CrossRef] - Xu, H.; Zhang, J.; Yang, J.; Lun, L. Node Importance Ranking of Complex Network based on Degree and Network Density. Int. J. Perform. Eng.
**2019**, 15, 850. [Google Scholar] [CrossRef] - Liu, J.; Xiong, Q.; Shi, W.; Shi, X.; Wang, K. Evaluating the importance of nodes in complex networks. Phys. A Stat. Mech. Its Appl.
**2016**, 452, 209–219. [Google Scholar] [CrossRef][Green Version] - Saxena, C.; Doja, M.N.; Ahmad, T. Group based centrality for immunization of complex networks. Phys. A Stat. Mech. Its Appl.
**2018**, 508, 35–47. [Google Scholar] [CrossRef][Green Version] - Magelinski, T.; Bartulovic, M.; Carley, K.M. Measuring Node Contribution to Community Structure with Modularity Vitality. IEEE Trans. Netw. Sci. Eng.
**2021**, 8, 707–723. [Google Scholar] [CrossRef] - Ghalmane, Z.; Hassouni, M.E.; Cherifi, H. Immunization of networks with non-overlapping community structure. Soc. Netw. Anal. Min.
**2019**, 9, 1–22. [Google Scholar] [CrossRef][Green Version] - Ghalmane, Z.; Cherifi, C.; Cherifi, H.; Hassouni, M.E. Centrality in complex networks with overlapping community structure. Sci. Rep.
**2019**, 9, 1–29. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gupta, N.; Singh, A.; Cherifi, H. Community-based immunization strategies for epidemic control. In Proceedings of the 2015 7th International Conference on Communication Systems and Networks (COMSNETS), Bangalore, India, 6–10 January 2015; pp. 1–6. [Google Scholar]
- Wang, J.W.; Rong, L.; Guo, T.Z. A new measure method of network node importance based on local characteristics. J. Dalian Univ. Technol.
**2010**, 50, 822–826. [Google Scholar] - Ren, Z.; Shao, F.; Liu, J.; Guo, Q.; Wang, B.H. Node importance measurement based on the degree and clustering coefficient information. Acta Phys. Sin.
**2013**, 62, 128901. [Google Scholar] - Yang, Y.; Yu, L.; Wang, X.; Zhou, Z.; Chen, Y.; Kou, T. A novel method to evaluate node importance in complex networks. Phys. A Stat. Mech. Its Appl.
**2019**, 526, 121118. [Google Scholar] [CrossRef] - Wang, J.; Wu, X.; Yan, B.; Guo, J. Improved method of node importance evaluation based on node contraction in complex networks. Procedia Eng.
**2011**, 15, 1600–1604. [Google Scholar] - Mo, H.; Deng, Y. Identifying node importance based on evidence theory in complex networks. Phys. A Stat. Mech. Its Appl.
**2019**, 529, 121538. [Google Scholar] [CrossRef] - Barrat, A.; Barthelemy, M.; Pastor-Satorras, R.; Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 3747–3752. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lü, L.; Zhou, T.; Zhang, Q.M.; Stanley, H.E. The H-index of a network node and its relation to degree and coreness. Nat. Commun.
**2016**, 7, 1–7. [Google Scholar] [CrossRef] [PubMed][Green Version] - Garas, A.; Schweitzer, F.; Havlin, S. A k-shell decomposition method for weighted networks. New J. Phys.
**2012**, 14, 083030. [Google Scholar] [CrossRef] - Wei, B.; Liu, J.; Wei, D.; Gao, C.; Deng, Y. Weighted k-shell decomposition for complex networks based on potential edge weights. Phys. A Stat. Mech. Its Appl.
**2015**, 420, 277–283. [Google Scholar] [CrossRef] - Newman, M. Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys. Rev. E
**2001**, 64, 016132. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brandes, U. A faster algorithm for betweenness centrality. J. Math. Sociol.
**2001**, 25, 163–177. [Google Scholar] [CrossRef] - Ou, Q.; Jin, Y.; Zhou, T.; Wang, B.H.; Yin, B.Q. Power-law strength-degree correlation from resource-allocation dynamics on weighted networks. Phys. Rev. E
**2007**, 75, 021102. [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] - Yang, Y.; Xie, G.; Xie, J. Mining important nodes in directed weighted complex networks. Discret. Dyn. Nat. Soc.
**2017**, 2017, 9741824. [Google Scholar] [CrossRef] - Qi, X.; Fuller, E.; Wu, Q.; Wu, Y.; Zhang, C.Q. Laplacian centrality: A new centrality measure for weighted networks. Inf. Sci.
**2012**, 194, 240–253. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ahmad, A.; Ahmad, T.; Bhatt, A. HWSMCB: A community-based hybrid approach for identifying influential nodes in the social network. Phys. A Stat. Mech. Its Appl.
**2020**, 545, 123590. [Google Scholar] [CrossRef] - Skibski, O.; Rahwan, T.; Michalak, T.; Yokoo, M. Attachment centrality: Measure for connectivity in networks. Artif. Intell.
**2019**, 274, 151–179. [Google Scholar] [CrossRef] - Sosnowska, J.; Skibski, O. Attachment Centrality for Weighted Graphs. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17), Melbourne, Australia, 19–25 August 2017; pp. 416–422. [Google Scholar]
- Chartrand, G.; Lesniak, L. Graphs and Digraphs, 1st ed.; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- West, D.B. Introduction to Graph Theory, 2nd ed.; Prentice Hall: Englewood Cliffs, NY, USA, 2001. [Google Scholar]
- Musial, K.; Juszczyszyn, K. Properties of bridge nodes in social networks. In International Conference on Computational Collective Intelligence; Springer: Berlin/Heidelberg, Germany, 2019; pp. 357–364. [Google Scholar]
- Zachary, W.W. An Information Flow Model for Conflict and Fission in Small Groups. J. Anthropol. Res.
**1977**, 33, 452–473. [Google Scholar] [CrossRef][Green Version] - Colizza, V.; Pastor-Satorras, R.; Vespignani, A. Reaction–diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys.
**2007**, 3, 276–282. [Google Scholar] [CrossRef][Green Version] - Mersch, D.P.; Crespi, A.; Keller, L. Tracking individuals shows spatial fidelity is a key regulator of ant social organization. Science
**2013**, 340, 1090–1093. [Google Scholar] [CrossRef][Green Version] - Firth, J.A.; Sheldon, B.C. Experimental manipulation of avian social structure reveals segregation is carried over across contexts. Proc. R. Soc. B Biol. Sci.
**2015**, 282, 20142350. [Google Scholar] [CrossRef][Green Version] - Rossi, R.; Ahmed, N.K. The Network Data Repository with Interactive Graph Analytics and Visualization. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, Austin, TX, USA, 25–30 January 2015. [Google Scholar]
- Latora, V.; Nicosia, V.; Russo, G. Complex Networks: Principles, Methods and Applications, 1st ed.; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Rubinov, M.; Sporns, O. Complex network measures of brain connectivity: Uses and interpretations. NeuroImage
**2010**, 52, 1059–1069. [Google Scholar] [CrossRef] - Latora, V.; Marchiori, M. Efficient Behavior of Small-World Networks. Phys. Rev. Lett.
**2001**, 87, 198701. [Google Scholar] [CrossRef][Green Version] - Lai, Y.; Motter, A.; Nishikawa, T. Attacks and cascades in complex networks. In Complex Networks; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2004; Volume 650. [Google Scholar]
- Sciarra, C.; Chiarotti, G.; Laio, F.; Ridolfi, L. A change of perspective in network centrality. Sci. Rep.
**2018**, 8, 1–9. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**The relation between decline rate of network efficiency and the ranking DIL-W${}^{1}$, DIL-W${}^{0.5}$ and DIL-W on the example networks.

**Figure 3.**The decline rate of the network efficiency as a function of deleting the top 10% of the vertices ranked by DIL-W${}^{1}$, DIL-W${}^{0.5}$ and DIL-W from five real networks (Zacharys karate club, wild birds, US air transport, brain functional coactivations and colony of ants).

**Figure 4.**The decline rate of the network efficiency as a function of deleting the top 10% of the vertices ranked by DIL-W${}^{0.1}$, DIL-W${}^{0.3}$, DIL-W${}^{0.5}$, DIL-W${}^{0.7}$, DIL-W${}^{0.9}$ and DIL-W${}^{1}$ from five real networks (Zacharys karate club, wild birds, US air transport, brain functional coactivations and colony of ants).

**Figure 5.**On the left the relationship between the DIL-W and DIL-W

^{α}rankings for different values of α. On the right the correlation coefficient for the different values of α.

**Table 1.**The ranking results of DIL-W and DIL-W${}^{1}$ on wild birds and US air transport networks.

Wild Birds Network | US Air Transport Network | |||
---|---|---|---|---|

DIL-W | DIL-W${}^{1}$ | DIL-W | DIL-W${}^{1}$ | |

Ranking Position | ID | ID | ID | ID |

1 | 12 | 12 | 1 | 1 |

2 | 27 | 27 | 3 | 3 |

3 | 23 | 23 | 6 | 6 |

4 | 10 | 10 | 10 | 10 |

5 | 35 | 35 | 14 | 14 |

6 | 77 | 77 | 7 | 5 |

7 | 25 | 25 | 5 | 7 |

8 | 51 | 51 | 4 | 4 |

9 | 26 | 26 | 8 | 8 |

10 | 94 | 94 | 2 | 2 |

11 | 6 | 6 | 12 | 12 |

12 | 56 | 56 | 11 | 11 |

13 | 61 | 61 | 13 | 13 |

14 | 11 | 11 | 21 | 21 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Manríquez, R.; Guerrero-Nancuante, C.; Martínez, F.; Taramasco, C. A Generalization of the Importance of Vertices for an Undirected Weighted Graph. *Symmetry* **2021**, *13*, 902.
https://doi.org/10.3390/sym13050902

**AMA Style**

Manríquez R, Guerrero-Nancuante C, Martínez F, Taramasco C. A Generalization of the Importance of Vertices for an Undirected Weighted Graph. *Symmetry*. 2021; 13(5):902.
https://doi.org/10.3390/sym13050902

**Chicago/Turabian Style**

Manríquez, Ronald, Camilo Guerrero-Nancuante, Felipe Martínez, and Carla Taramasco. 2021. "A Generalization of the Importance of Vertices for an Undirected Weighted Graph" *Symmetry* 13, no. 5: 902.
https://doi.org/10.3390/sym13050902