Abstract
Graph robustness upon node failures state-of-art is huge. However, not enough is known on the effects of centrality metrics ranking after graph perturbations. To fill this gap, our aim is to quantify how much small graph perturbations will affect the centrality metrics. Thus, we considered two type of probabilistic failure models (i.e., Uniform and Best Connected), a fraction \(\tau \) of nodes under attack, with \(0 < \tau \le 1\), and three popular centrality metrics (i.e., Degree, the Eigenvector and the Katz centrality). We discovered that in the Uniform model the amount of change in the adjacency matrix due to a perturbation is not significantly affected when \(\tau \) is small even with a quite high failure probability (i.e., \(p\le 85\%\)) and that the Eigenvector centrality is the most susceptible metric to deformation respect to the others herein analysed; whereas, in the Best Connected model, the amount of perturbation is proportional to \(\tau \).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
A walk of length k in a graph is a sequence \(\langle i_0, i_1,\ldots ,i_{k-1} \rangle \) of nodes such that pair of consecutive nodes in the sequence are connected by an edge.
References
Newman, M.: Networks: An Introduction. Oxford University Press (2010)
Lü, L., Chen, D., Ren, X., Zhang, Q., Zhang, Y., Zhou, T.: Vital nodes identification in complex networks. Phys. Rep. 650, 1–63 (2016)
Kang, C., Kraus, S., Molinaro, C., Spezzano, F., Subrahmanian, V.: Diffusion centrality: a paradigm to maximize spread in social networks. Artif. Intell. 239, 70–96 (2016)
Albert, R., Jeong, H., Barabási, A.: Error and attack tolerance of complex networks. Nature 406(6794), 378–382 (2000)
Callaway, D.S., Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85(25), 5468 (2000)
Li, M., Liu, R., Lü, L., Hu, M., Xu, S., Zhang, Y.: Percolation on complex networks: theory and application. Phys. Rep. 907, 1–68 (2021)
Moore, C., Newman, M.: Exact solution of site and bond percolation on small-world networks. Phys. Rev. E 62(5), 7059 (2000)
Cavallaro, L., Costantini, S., De Meo, P., Liotta, A., Stilo, G.: Network connectivity under a probabilistic node failure model. In: IEEE Transactions on Network Science and Engineering, pp. 1–1 (2022)
Barabási, A., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Costenbader, E., Valente, T.W.: The stability of centrality measures when networks are sampled. Soc. Netw. 25(4), 283–307 (2003)
Borgatti, S.P., Carley, M.M., Krackhardt, D.: On the robustness of centrality measures under conditions of imperfect data. Soc. Netw. 28(2), 124–136 (2006)
Diesner, J., Evans, C.S., Kim, J.: Impact of entity disambiguation errors on social network properties. In: Ninth International AAAI Conference on Web and Social Media (2015)
Mishra, S., Fegley, B.D., Diesner, J., Torvik, V.I.: Self-citation is the hallmark of productive authors, of any gender. PloS One 13(9), e0195773 (2018)
Cvetković, D., Rowlinson, P., Simic, S.: Eigenspaces of Graphs, vol. 66. Cambridge University Press (1997)
Strang, G.: Introduction to Linear Algebra, vol. 3. Cambridge Press Wellesley, Wellesley, MA (1993)
Langville, A., Meyer, C.: Google’s PageRank and Beyond. Princeton university press (2011)
Benzi, M., Klymko, C.: On the limiting behavior of parameter-dependent network centrality measures. SIAM J. Matrix Anal. Appl. 36(2), 686–706 (2015)
Franklin, J.: Matrix Theory. Courier Corporation (2012)
Rozemberczki, B., Allen, C., Sarkar, R.: Multi-scale attributed node embedding. J. Complex Netw. 9(2):cnab014 (2021)
Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: densification and shrinking diameters. ACM Trans. Knowl. Disc. Data (TKDD) 1(1):2–es (2007)
Grando, F., Zambenedetti Granville, L., Lamb, L.C.: Machine learning in network centrality measures: tutorial and outlook. ACM Comput. Surv. 51(5):102:1–102:32 (2019)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Cavallaro, L. et al. (2023). Analysis on the Effects of Graph Perturbations on Centrality Metrics. In: Cherifi, H., Mantegna, R.N., Rocha, L.M., Cherifi, C., Micciche, S. (eds) Complex Networks and Their Applications XI. COMPLEX NETWORKS 2016 2022. Studies in Computational Intelligence, vol 1078. Springer, Cham. https://doi.org/10.1007/978-3-031-21131-7_34
Download citation
DOI: https://doi.org/10.1007/978-3-031-21131-7_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-21130-0
Online ISBN: 978-3-031-21131-7
eBook Packages: EngineeringEngineering (R0)