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On the impact of network size and average degree on the robustness of centrality measures

Published online by Cambridge University Press:  20 October 2020

Christoph Martin Open the ORCID record for Christoph Martin [Opens in a new window]  and
Peter Niemeyer
Christoph Martin*
Affiliation:
Institute of Information Systems, Leuphana University of Lüneburg 21335 Lüneburg, Germany (e-mail: [email protected])
Peter Niemeyer
Affiliation:
Institute of Information Systems, Leuphana University of Lüneburg 21335 Lüneburg, Germany (e-mail: [email protected])
*
*Corresponding author. Email: [email protected]
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Abstract

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Measurement errors are omnipresent in network data. Most studies observe an erroneous network instead of the desired error-free network. It is well known that such errors can have a severe impact on network metrics, especially on centrality measures: a central node in the observed network might be less central in the underlying, error-free network. The robustness is a common concept to measure these effects. Studies have shown that the robustness primarily depends on the centrality measure, the type of error (e.g., missing edges or missing nodes), and the network topology (e.g., tree-like, core-periphery). Previous findings regarding the influence of network size on the robustness are, however, inconclusive. We present empirical evidence and analytical arguments indicating that there exist arbitrary large robust and non-robust networks and that the average degree is well suited to explain the robustness. We demonstrate that networks with a higher average degree are often more robust. For the degree centrality and Erdős–Rényi (ER) graphs, we present explicit formulas for the computation of the robustness, mainly based on the joint distribution of node degrees and degree changes which allow us to analyze the robustness for ER graphs with a constant average degree or increasing average degree.

Keywords

centrality measuresmeasurement errormissing datarobustness
Type
Research Article
Information
Network Science , Volume 9 , Special Issue S1: Complex Networks 2019 , October 2021 , pp. S61 - S82
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

Special Issue Editor: Hocine Cherifi

References

Albert, R., Jeong, H., & Barabási, A.-L. (2000). Error and attack tolerance of complex networks. Nature, 406(July), 378–382.CrossRefGoogle ScholarPubMed
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(October), 509–512.CrossRefGoogle ScholarPubMed
Beuming, T., Skrabanek, L., Niv, M. Y., Mukherjee, P., & Weinstein, H. (2005). PDZBase: A protein–protein interaction database for PDZ-domains. Bioinformatics, 21(6), 827–828.CrossRefGoogle ScholarPubMed
Boguñá, M., Pastor-Satorras, R., Díaz-Guilera, A., & Arenas, A. (2004). Models of social networks based on social distance attachment. Physical Review E, 70(5), 056122.CrossRefGoogle ScholarPubMed
Bolland, J. M. (1988). Sorting out centrality: An analysis of the performance of four centrality models in real and simulated networks. Social Networks, 10(3), 233–253.CrossRefGoogle Scholar
Bonacich, Phillip. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170–1182.CrossRefGoogle Scholar
Borgatti, S. P., Carley, K. M., & Krackhardt, D. (2006). On the robustness of centrality measures under conditions of imperfect data. Social Networks, 28(2), 124–136.CrossRefGoogle Scholar
Brandes, U. (2001). A faster algorithm for betweenness centrality*. The Journal of Mathematical Sociology, 25(2), 163–177.CrossRefGoogle Scholar
Brin, S., & Page, L. (1998). The anatomy of a large-scale hypertextual web search engine. In Seventh international world-wide web conference (WWW 1998).CrossRefGoogle Scholar
Callaway, D. S., Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2000). Network robustness and fragility: Percolation on random graphs. Physical Review Letters, 85(25), 5468–5471.CrossRefGoogle ScholarPubMed
Cho, E., Myers, S. A., & Leskovec, J. (2011). Friendship and mobility: User movement in location-based social networks. In Proceedings of the international conference on knowledge discovery and data mining (pp. 1082–1090).CrossRefGoogle Scholar
Costenbader, E., & Valente, T. W. (2003). The stability of centrality measures when networks are sampled. Social Networks, 25(4), 283–307.CrossRefGoogle Scholar
De Las Rivas, J., & Fontanillo, C. (2010). Protein-protein interactions essentials: Key concepts to building and analyzing interactome networks. Plos Computational Biology, 6(6), e1000807.CrossRefGoogle ScholarPubMed
Dünker, D., & Kunegis, J. (2015). Social networking by proxy: Analysis of Dogster, Catster and Hamsterster. In Proceedings of international conference on world wide web companion (pp. 361–362).CrossRefGoogle Scholar
Erdös, P., & Rényi, A. (1959). On random graphs. Publicationes Mathematicae, 6, 290–297.Google Scholar
Erman, N., & Todorovski, L. (2015). The effects of measurement error in case of scientific network analysis. Scientometrics, 104(2), 453–473.CrossRefGoogle Scholar
Frantz, T. L., Cataldo, M., & Carley, K. M. (2009). Robustness of centrality measures under uncertainty: Examining the role of network topology. Computational and Mathematical Organization Theory, 15(4), 303–328.CrossRefGoogle Scholar
Ghoshal, G., & Barabási, A.-L. (2011). Ranking stability and super-stable nodes in complex networks. Nature Communications, 2, 394.CrossRefGoogle ScholarPubMed
Gleiser, P. M., & Danon, L. (2003). Community Structure in Jazz. Advances in Complex Systems, 6(4), 565–573.CrossRefGoogle Scholar
Goodman, L. A, & Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49(268), 732–764.Google Scholar
Hagberg, A. A., Schult, D. A., & Swart, P. J. (2008). Exploring network structure, dynamics, and function using NetworkX. In Proceedings of the 7th python in science conference (SciPy2008) (pp. 11–15).Google Scholar
Holzmann, H., Anand, A., & Khosla, M. (2019). Delusive pagerank in incomplete graphs. In L. M. Aiello, C. Cherifi, H. Cherifi, R. Lambiotte, P. Lió, & L. M. Rocha (Eds.), Complex networks and their applications VII (pp. 104–117). Cham: Springer International Publishing.Google Scholar
Joshi-Tope, G., Gillespie, M., Vastrik, I., D’Eustachio, P., Schmidt, E., de Bono, B., Jassal, B., Gopinath, G. R., Wu, G. R., Matthews, L., et al. (2005). Reactome: A knowledgebase of biological pathways. Nucleic Acids Research, 33(Suppl. 1), D428–D432.Kendall, M G. (1945). The treatment of ties in ranking problems. Biometrika, 33(3), 239–251.Google Scholar
Kim, P. J., & Jeong, H. (2007). Reliability of rank order in sampled networks. European Physical Journal B, 55(1), 109–114.CrossRefGoogle Scholar
Koschützki, D., Lehmann, K. A., & Peeters, L. (2005). Centrality indices. In U. Brandes, & T. Erlebach (Eds.), Network analysis: methodological foundations (pp. 16–61). Springer Berlin Heidelberg.Google Scholar
Kossinets, Gueorgi. (2006). Effects of missing data in social networks. Social networks, 28(3), 247–268.CrossRefGoogle Scholar
Kunegis, J. (2013). KONECT - The koblenz network collection. In WWW 2013 companion – Proceedings of the 22nd international conference on World Wide Web.Google Scholar
Leskovec, J., & Mcauley, J. J. (2012). Learning to discover social circles in ego networks. In F. Pereira, C. J. C. Burges, L. Bottou, & K. Q. Weinberger (Eds.), Advances in neural information processing systems 25 (pp. 539–547). Curran Associates, Inc.Google Scholar
Leskovec, J., Kleinberg, J., & Faloutsos, C. (2007). Graph evolution: Densification and shrinking diameters. The ACM Transactions on Knowledge Discovery from Data, 1(1).CrossRefGoogle Scholar
Lusseau, D., Schneider, K., Boisseau, O. J., Haase, P., Slooten, E., & Dawson, S. M. (2003). The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations. Behavioral Ecology and Sociobiology, 54, 396–405.CrossRefGoogle Scholar
Marsden, P. V. (1990). Network data and measurement. Annual Review of Sociology, 16(1), 435–463.CrossRefGoogle Scholar
Martin, C., & Niemeyer, P. (2019). Influence of measurement errors on networks: Estimating the robustness of centrality measures. Network Science, 7(2), 180–195.CrossRefGoogle Scholar
Martin, C., & Niemeyer, P. (2020). The role of network size for the robustness of centrality measures. In H. Cherifi, S. Gaito, J. F. Mendes, E. Moro, & L. M. Rocha (Eds.), Complex networks and their applications VIII (pp. 40–51). Cham: Springer International Publishing.Google Scholar
Murai, S., & Yoshida, Y. (2019). Sensitivity analysis of centralities on unweighted networks. In The world wide web conference. WWW 2019 (pp. 1332–1342). New York, NY, USA: ACM.Google Scholar
Newman, M. (2003). The structure and function of complex networks. Siam Review, 45(2), 167–256.CrossRefGoogle Scholar
Newman, M. E., Strogatz, S. H., & Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 64.CrossRefGoogle Scholar
Niu, Q., Zeng, A., Fan, Y., & Di, Z. (2015). Robustness of centrality measures against network manipulation. Physica A: Statistical Mechanics and Its Applications, 438, 124–131.CrossRefGoogle Scholar
Platig, J., Ott, E., & Girvan, M. (2013). Robustness of network measures to link errors. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 88(6).CrossRefGoogle Scholar
Rozemberczki, B., Davies, R., Sarkar, R., & Sutton, C. (2019). GEMSEC: Graph embedding with self clustering. Proceedings of the 2019 IEEE/ACM international conference on advances in social networks analysis and mining 2019 (pp. 65–72). ACM.CrossRefGoogle Scholar
Rual, J.-F., Venkatesan, K., Hao, T., Hirozane-Kishikawa, T., Dricot, A., Li, N., Berriz, G. F., Gibbons, F. D., Dreze, M., & Ayivi-Guedehoussou, N. (2005). Towards a proteome-scale map of the human protein–protein interaction network. Nature, 1173–1178.CrossRefGoogle Scholar
Schulz, J. (2016). Using Monte Carlo simulations to assess the impact of author name disambiguation quality on different bibliometric analyses. Scientometrics, 107(3), 1283–1298.CrossRefGoogle Scholar
Smith, J. A., & Moody, J. (2013). Structural effects of network sampling coverage I: Nodes missing at random. Social Networks, 35(4).CrossRefGoogle Scholar
Smith, J. A., Moody, J., & Morgan, J. H. (2017). Network sampling coverage II: The effect of non-random missing data on network measurement. Social Networks, 48, 78–99.CrossRefGoogle ScholarPubMed
Tsugawa, S., & Ohsaki, H. (2015). Analysis of the robustness of degree centrality against random errors in graphs. In Studies in computational intelligence, vol. 597 (pp. 25–36).CrossRefGoogle Scholar
Wang, C., Butts, C. T., Hipp, J. R., Jose, R., & Lakon, C. M. (2016). Multiple imputation for missing edge data: A predictive evaluation method with application to Add Health. Social Networks, 45, 89–98.CrossRefGoogle ScholarPubMed
Wang, D. J., Shi, X., McFarland, D. A., & Leskovec, J. (2012). Measurement error in network data: A re-classification. Social Networks, 34(4), 396–409.CrossRefGoogle Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393(1), 440–442.CrossRefGoogle ScholarPubMed
Yang, J., & Leskovec, J. (2012). Defining and evaluating network communities based on ground-truth. In Proceedings of ACM SIGKDD workshop on mining data semantics (p. 3). ACM.CrossRefGoogle Scholar
Zachary, W. (1977). An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33, 452–473.CrossRefGoogle Scholar
Zafarani, R., & Liu, H. (2009). Social computing data repository at ASU.Google Scholar