The Role of Network Size for the Robustness of Centrality Measures

  • Christoph MartinEmail author
  • Peter Niemeyer
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 881)


Measurement errors are omnipresent in network data. Studies have shown that these errors have a severe impact on the robustness of centrality measures. It has been observed that the robustness mainly depends on the network structure, the centrality measure, and the type of error. Previous findings regarding the influence of network size on robustness are, however, inconclusive.

Based on twenty-four empirical networks, we investigate the relationship between global network measures, especially network size and average degree, and the robustness of the degree, eigenvector centrality, and PageRank. We demonstrate that, in the vast majority of cases, networks with a higher average degree are more robust.

For random graphs, we observe that the robustness of Erdős-Rényi (ER) networks decreases with an increasing average degree, whereas with Barabàsi-Albert networks, the opposite effect occurs: with an increasing average degree, the robustness also increases.

As a first step into an analytical discussion, we prove that for ER networks of different size but with the same average degree, the robustness of the degree centrality remains stable.


Centrality Robustness Measurement error Missing data Noisy data Sampling 


  1. 1.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonacich, P.: Power and centrality: a family of measures. Am. J. Sociol. 92(5), 1170–1182 (1987)CrossRefGoogle Scholar
  3. 3.
    Borgatti, S.P., Carley, K.M., Krackhardt, D.: On the robustness of centrality measures under conditions of imperfect data. Soc. Netw. 28(2), 124–136 (2006)CrossRefGoogle Scholar
  4. 4.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. In: Seventh International World-Wide Web Conference (WWW 1998) (1998)Google Scholar
  5. 5.
    Costenbader, E., Valente, T.W.: The stability of centrality measures when networks are sampled. Soc. Netw. 25(4), 283–307 (2003)CrossRefGoogle Scholar
  6. 6.
    De Las Rivas, J., Fontanillo, C.: Protein-protein interactions essentials: key concepts to building and analyzing interactome networks. PLoS Comput. Biol. 6(6), e1000807 (2010)CrossRefGoogle Scholar
  7. 7.
    Erdös, P., Rényi, A.: On random graphs. Publicationes Mathematicae 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erman, N., Todorovski, L.: The effects of measurement error in case of scientific network analysis. Scientometrics 104(2), 453–473 (2015)CrossRefGoogle Scholar
  9. 9.
    Frantz, T.L., Cataldo, M., Carley, K.M.: Robustness of centrality measures under uncertainty: examining the role of network topology. Comput. Math. Organ. Theory 15(4), 303–328 (2009)CrossRefGoogle Scholar
  10. 10.
    Ghoshal, G., Barabási, A.-L.: Ranking stability and super-stable nodes in complex networks. Nat. Commun. 2, 394 (2011)CrossRefGoogle Scholar
  11. 11.
    Goodman, L.A., Kruskal, W.H.: Measures of association for cross classifications. J. Am. Stat. Assoc. 49(268), 732–764 (1954)zbMATHGoogle Scholar
  12. 12.
    Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 11–15 (2008)Google Scholar
  13. 13.
    Holzmann, H., Anand, A., Khosla, M.: Delusive PageRank in incomplete graphs. In: Aiello, L.M., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L.M. (eds.) Complex Networks and Their Applications VII, pp. 104–117. Springer, Cham (2019)Google Scholar
  14. 14.
    Kendall, M.G.: The treatment of ties in ranking problems. Biometrika 33(3), 239–251 (1945)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kim, P.J., Jeong, H.: Reliability of rank order in sampled networks. Eur. Phys. J. B 55(1), 109–114 (2007)CrossRefGoogle Scholar
  16. 16.
    Koschützki, D., Lehmann, K., Peeters, L.: Centrality indices. In: Brandes, U., Erlebach, T. (eds.) Network Analysis: Methodological Foundations, pp. 16–61. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Kossinets, G.: Effects of missing data in social networks. Soc. Netw. 28(3), 247–268 (2006)CrossRefGoogle Scholar
  18. 18.
    Kunegis, J.: KONECT - the Koblenz network collection. In: WWW 2013 Companion - Proceedings of the 22nd International Conference on World Wide Web (2013)Google Scholar
  19. 19.
    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: densification and shrinking diameters. ACM Trans. Knowl. Discov. Data 1, 1 (2007)CrossRefGoogle Scholar
  20. 20.
    Marsden, P.: Network data and measurement. Annu. Rev. Sociol 16(1), 435–463 (1990)CrossRefGoogle Scholar
  21. 21.
    Martin, C., Niemeyer, P.: Influence of measurement errors on networks: estimating the robustness of centrality measures. Netw. Sci. 7(2), 180–195 (2019)CrossRefGoogle Scholar
  22. 22.
    Murai, S., Yoshida, Y.: Sensitivity analysis of centralities on unweighted networks. In: The World Wide Web Conference, WWW 2019, New York, NY, USA, pp. 1332–1342. ACM (2019)Google Scholar
  23. 23.
    Newman, M.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Niu, Q., Zeng, A., Fan, Y., Di, Z.: Robustness of centrality measures against network manipulation. Phys. A 438, 124–131 (2015)CrossRefGoogle Scholar
  25. 25.
    Platig, J., Ott, E., Girvan, M.: Robustness of network measures to link errors. Phys. Rev. E-Stat. Nonlinear Soft Matter Phys. 88, 6 (2013)CrossRefGoogle Scholar
  26. 26.
    Schulz, J.: Using Monte Carlo simulations to assess the impact of author name disambiguation quality on different bibliometric analyses. Scientometrics 107(3), 1283–1298 (2016)CrossRefGoogle Scholar
  27. 27.
    Smith, J.A., Moody, J.: Structural effects of network sampling coverage I: nodes missing at random. Soc. Netw. 35, 4 (2013)CrossRefGoogle Scholar
  28. 28.
    Smith, J.A., Moody, J., Morgan, J.H.: Network sampling coverage II: the effect of non-random missing data on network measurement. Soc. Netw. 48, 78–99 (2017)CrossRefGoogle Scholar
  29. 29.
    Tsugawa, S., Ohsaki, H.: Analysis of the robustness of degree centrality against random errors in graphs. Studies in Computational Intelligence, vol. 597, pp. 25–36 (2015)Google Scholar
  30. 30.
    Wang, C., Butts, C.T., Hipp, J.R., Jose, R., Lakon, C.M.: Multiple imputation for missing edge data: a predictive evaluation method with application to Add Health. Soc. Netw. 45, 89–98 (2016)CrossRefGoogle Scholar
  31. 31.
    Wang, D.J., Shi, X., McFarland, D.A., Leskovec, J.: Measurement error in network data: a re-classification. Soc. Netw. 34(4), 396–409 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Information SystemsLeuphana University of LüneburgLüneburgGermany

Personalised recommendations