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Re-conceptualizing centrality in social networks†

Published online by Cambridge University Press:  26 September 2016

D. SCHOCH  and
U. BRANDES
D. SCHOCH
Affiliation:
Departement of Computer & Information Science, University of Konstanz, Konstanz, Germany emails: david.schoch@uni-konstanz.de, ulrik.brandes@uni-konstanz.de Graduate School of Decision Sciences, University of Konstanz, Konstanz, Germany
U. BRANDES
Affiliation:
Departement of Computer & Information Science, University of Konstanz, Konstanz, Germany emails: david.schoch@uni-konstanz.de, ulrik.brandes@uni-konstanz.de Graduate School of Decision Sciences, University of Konstanz, Konstanz, Germany
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Abstract

In the social sciences, networks are used to represent relationships between social actors, be they individuals or aggregates. The structural importance of these actors is assessed in terms of centrality indices which are commonly defined as graph invariants. Many such indices have been proposed, but there is no unifying theory of centrality. Previous attempts at axiomatic characterization have been focused on particular indices, and the conceptual frameworks that have been proposed alternatively do not lend themselves to mathematical treatment.

We show that standard centrality indices, although seemingly distinct, can in fact be expressed in a common framework based on path algebras. Since, as a consequence, all of these indices preserve the neighbourhood-inclusion pre-order, the latter provides a conceptually clear criterion for the definition of centrality indices.

Keywords

network sciencesocial networkscentralityrankingpositional dominance
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Special Issue 6: Network Analysis and Modelling , December 2016 , pp. 971 - 985
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

†

We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft under grant Br 2158/6-1. Part of this research was presented at the SIAM Workshop on Network Science (Snowbird, Utah, May 2015).

References

REFERENCES

[1] Batagelj, V. (1994) Semirings for social network analysis. J. Math. Sociol. 19 (1), 53–68.CrossRefGoogle Scholar
[2] Bavelas, A. (1948) A mathematical model for group structures. Human Organizations 7 (3), 16–30.Google Scholar
[3] Bavelas, A. (1950) Communication patterns in task-oriented groups. J. Acoust. Soc. Am. 22 (6), 725–730.CrossRefGoogle Scholar
[4] Beauchamp, M. A. (1965) An improved index of centrality. Behav. Sci. 10, 161–163.Google Scholar
[5] Benzi, M. & Klymko, C. (2013) Total communicability as a centrality measure. J. Complex Netw. 1 (2), 124–149.Google Scholar
[6] Bloch, F., Jackson, M. O. & Tebaldi, P. (2016) Centrality measures in networks. SSRN Electron. J.. Available at: http://dx.doi.org/10.2139/ssrn.2749124.Google Scholar
[7] Boldi, P. & Vigna, S. (2014) Axioms for centrality. Internet Math. 10 (3–4), 222–262.Google Scholar
[8] Bonacich, P. (1972) Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 2, 113–120.CrossRefGoogle Scholar
[9] Bonacich, P. (1987) Power and centrality: A family of measures. Am. J. Sociol. 92 (5), 1170–1182.Google Scholar
[10] Borgatti, S. P. (2005) Centrality and network flow. Soc. Netw. 27 (1), 55–71.Google Scholar
[11] Borgatti, S. P., Carley, K. M. & Krackhardt, D. (2006) On the robustness of centrality measures under conditions of imperfect data. Soc. Netw. 28 (2), 124–136.Google Scholar
[12] Borgatti, S. P. & Everett, M. G. (2006) A graph-theoretic perspective on centrality. Soc. Netw. 28 (4), 466–484.CrossRefGoogle Scholar
[13] Borgatti, S. P., Everett, M. G. & Johnson, J. C. (2013) Analyzing Social Networks. Sage, London.Google Scholar
[14] Botafogo, R. A., Rivlin, E. & Shneiderman, B. (1992) Structural analysis of hypertexts: Identifying hierarchies and useful metrics. ACM Trans. Inform. Syst. 10 (2), 142–180.Google Scholar
[15] Brandes, U. (2016) Network positions. Methodological Innov. 9, 2059799116630650.Google Scholar
[16] Brandes, U. & Erlebach, T. (editors) (2005) Network Analysis: Methodological Foundations, Lecture Notes in Computer Science, Vol. 3418, Springer-Verlag, Berlin.Google Scholar
[17] Brandes, U., Robins, G., McCranie, A. & Wasserman, S. (2013) What is network science? Netw. Sci. 1 (1), 1–15.Google Scholar
[18] Costenbader, E. & Valente, T. W. (2003) The stability of centrality measures when networks are sampled. Soc. Netw. 25 (4), 283–307.Google Scholar
[19] Estrada, E. & Rodríguez-Velázquez, J. A. (2005) Subgraph centrality in complex networks. Phys. Rev. E 71 (5), 056103.CrossRefGoogle ScholarPubMed
[20] Foldes, S. & Hammer, P. L. (1978) The Dilworth number of a graph. Ann. Discrete Math. 2, 211–219.CrossRefGoogle Scholar
[21] Freeman, L. C. (1977) A set of measures of centrality based on betweenness. Sociometry 40 (1), 35–41.CrossRefGoogle Scholar
[22] Freeman, L. C. (1979) Centrality in social networks: Conceptual clarification. Soc. Netw. 1 (3), 215–239.Google Scholar
[23] Freeman, L. C. (2004) The Development of Social Network Analysis: A Study in the Sociology of Science, Empirical Press, Vancouver, BC.Google Scholar
[24] Gondran, M. & Minoux, M. (2008) Graphs, Diods and Semirings. Springer-Verlag, Berlin.Google Scholar
[25] Guimera, R., Mossa, S., Turtschi, A. & Nunes Amaral, L. A. (2005) The worldwide air transportation network: Anomalous centrality, community structure, and cities' global roles. Proc. Nat. Acad. Sci. USA 102 (22), 7794–7799.Google Scholar
[26] Hage, P. & Harary, F. (1995) Eccentricity and centrality in networks. Soc. Netw. 17, 57–63.CrossRefGoogle Scholar
[27] Hennig, M., Brandes, U., Pfeffer, J. & Mergel, I. (2012) Studying Social Networks – A Guide to Empirical Research. Campus, Frankfurt/New York.Google Scholar
[28] Junker, B. H., Koschützki, D. & Schreiber, F. (2006) Exploration of biological network centralities with CentiBiN. BMC Bioinformatics 7 (219).Google Scholar
[29] Kadushin, C. (2011) Understanding Social Networks: Theories, Concepts, and Findings. Oxford University Press, New York, NY.Google Scholar
[30] Katz, L. (1953) A new status index derived from sociometric analysis. Psychometrika 18 (1), 39–43.Google Scholar
[31] Kitti, M. (2016) Axioms for centrality scoring with principal eigenvectors. Soc. Choice Welf. 46 (3), 639–653.Google Scholar
[32] Koschützki, D., Lehmann, K. A., Peeters, L., Richter, S., Tenfelde-Podehl, D. & Zlotowski, O. (2005) Centrality indices. In: Brandes, U. & Erlebach, T. (editors), Network Analysis: Methodological Foundations, Lecture Notes in Computer Science, Vol. 3418, Springer-Verlag, Berlin, pp. 16–61.Google Scholar
[33] Koschützki, D., Lehmann, K. A., Tenfelde-Podehl, D. & Zlotowski, O. (2005) Advanced centrality concepts. In: Brandes, U. & Erlebach, T. (editors), Network Analysis: Methodological Foundations, Lecture Notes in Computer Science, Vol. 3418, Springer-Verlag, Berlin, pp. 83–111.Google Scholar
[34] Landherr, A., Friedl, B. & Heidemann, J. (2010) A critical review of centrality measures in social networks. Bus. Inform. Syst. Eng. 2 (6), 371–385.CrossRefGoogle Scholar
[35] Leavitt, H. J. (1951) Some effects of certain communication patterns on group performance. J. Abnormal Soc. Psychol. 46 (1), 38.CrossRefGoogle ScholarPubMed
[36] Mahadev, N. V. R. & Peled, U. N. (1995) Threshold Graphs and Related Topics, Annals of Discrete Mathematics, Vol. 56, North Holland, Amsterdam.Google Scholar
[37] Moreno, J. L. (1953) Who Shall Survive? Foundations of Sociometry, Group Psychotherapy and Sociodrama. Beacon House, New York, NY. First published in 1934.Google Scholar
[38] Neal, Z. (2013) A computationally efficient approximation of beta centrality. Connections 33 (1), 11–17.Google Scholar
[39] Nieminen, J. (1973) On the centrality in a directed graph. Soc. Sci. Res. 2 (4), 371–378.Google Scholar
[40] Nieminen, J. (1974) On the centrality in a graph. Scand. J. Psychol. 15, 332–336.Google Scholar
[41] Ruhnau, B. (2000) Eigenvector-centrality–a node-centrality? Soc. Netw. 22 (4), 357–365.Google Scholar
[42] Sabidussi, G. (1966) The centrality index of a graph. Psychometrika 31 (4), 581–603.Google Scholar
[43] Srivastava, H. M. & Manocha, H. L. (1984) A Treatise on Generating Functions. John Wiley and Sons, New York.Google Scholar
[44] Todeschini, R. & Consonni, V. (2009) Molecular Descriptors for Chemoinformatics, 2nd ed., Wiley-VCH, Weinheim.Google Scholar
[45] Tutzauer, F. (2007) Entropy as a measure of centrality in networks characterized by path-transfer flow. Soc. Netw. 29 (2), 249–265.Google Scholar
[46] Valente, T. W. & Foreman, R. K. (1998) Integration and radiality: Measuring the extent of an individual's connectedness and reachability in a network. Soc. Netw. 20 (1), 89–105.Google Scholar
[47] van den Brink, R. & Gilles, R. P. (2000) Measuring domination in directed networks. Soc. Netw. 22 (2), 141–157.CrossRefGoogle Scholar
[48] Wasserman, S. & Faust, K. (1994) Social Network Aanalysis. Methods and Applications, Cambridge University Press, Cambridge, UK.Google Scholar