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Central European Journal of Operations Research

, Volume 25, Issue 4, pp 771–790 | Cite as

Measuring centrality by a generalization of degree

  • László Csató
Original Paper

Abstract

Network analysis has emerged as a key technique in communication studies, economics, geography, history and sociology, among others. A fundamental issue is how to identify key nodes, for which purpose a number of centrality measures have been developed. This paper proposes a new parametric family of centrality measures called generalized degree. It is based on the idea that a relationship to a more interconnected node contributes to centrality in a greater extent than a connection to a less central one. Generalized degree improves on degree by redistributing its sum over the network with the consideration of the global structure. Application of the measure is supported by a set of basic properties. A sufficient condition is given for generalized degree to be rank monotonic, excluding counter-intuitive changes in the centrality ranking after certain modifications of the network. The measure has a graph interpretation and can be calculated iteratively. Generalized degree is recommended to apply besides degree since it preserves most favourable attributes of degree, but better reflects the role of the nodes in the network and has an increased ability to distinguish among their importance.

Keywords

Network Centrality measure Degree Axiomatic approach 

Notes

Acknowledgments

We are grateful to Herbert Hamers for drawing our attention to centrality measures, and to Dezső Bednay, Pavel Chebotarev and Tamás Sebestyén for useful advices. The research was supported by OTKA grant K 111797 and MTA-SYLFF (The Ryoichi Sasakawa Young Leaders Fellowship Fund) Grant ‘Mathematical analysis of centrality measures’, awarded to the author in 2015.

References

  1. Avrachenkov KE, Mazalov VV, Tsynguev BT (2015) Beta current flow centrality for weighted networks. In: Thai MT, Nguyen NP, Shen H (eds) Computational social networks, of lecture notes in computer science, vol 9197. Springer, New YorkGoogle Scholar
  2. Bavelas A (1948) A mathematical model for group structures. Hum Organ 7(3):16–30CrossRefGoogle Scholar
  3. Boldi P, Vigna S (2014) Axioms for centrality. Internet Math 10(3–4):222–262CrossRefGoogle Scholar
  4. Bonacich P (1987) Power and centrality: a family of measures. Am J Sociol 92(1):1170–1182CrossRefGoogle Scholar
  5. Borgatti SP, Everett MG (2006) A graph-theoretic perspective on centrality. Soc Netw 28(4):466–484CrossRefGoogle Scholar
  6. Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw ISDN Syst 30(1):107–117CrossRefGoogle Scholar
  7. Chebotarev P (1989) Generalization of the row sum method for incomplete paired comparisons. Autom Remote Control 50(8):1103–1113Google Scholar
  8. Chebotarev P (1994) Aggregation of preferences by the generalized row sum method. Math Soc Sci 27(3):293–320CrossRefGoogle Scholar
  9. Chebotarev P (2012) The walk distances in graphs. Discrete Appl Math 160(10):1484–1500CrossRefGoogle Scholar
  10. Chebotarev P, Shamis E (1997) The matrix-forest theorem and measuring relations in small social groups. Autom Remote Control 58(9):1505–1514Google Scholar
  11. Chien S, Dwork C, Kumar R, Simon DR, Sivakumar D (2004) Link evolution: analysis and algorithms. Internet Math 1(3):277–304CrossRefGoogle Scholar
  12. Csató L (2015) A graph interpretation of the least squares ranking method. Soc Choice Welf 44(1):51–69CrossRefGoogle Scholar
  13. Dequiedt V, Zenou Y (2015) Local and consistent centrality measures in networks. https://www.gate.cnrs.fr/IMG/pdf/Dequiedt2015.pdf
  14. Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40(1):35–41CrossRefGoogle Scholar
  15. Freeman LC (1979) Centrality in social networks: conceptual clarification. Soc Netw 1(3):215–239CrossRefGoogle Scholar
  16. Garg M (2009) Axiomatic foundations of centrality in networks. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1372441
  17. Geršgorin S (1931) Über die Abgrenzung der Eigenwerte einer Matrix. Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et naturelles 6:749–754Google Scholar
  18. González-Díaz J, Hendrickx R, Lohmann E (2014) Paired comparisons analysis: an axiomatic approach to ranking methods. Soc Choice Welf 42(1):139–169CrossRefGoogle Scholar
  19. Jackson MO (2010) Social and economic networks. Princeton University Press, PrincetonGoogle Scholar
  20. Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43CrossRefGoogle Scholar
  21. Kitti M (2012) Axioms for centrality scoring with principal eigenvectors. http://www.ace-economics.fi/kuvat/dp79.pdf
  22. Klein DJ (2010) Centrality measure in graphs. J Math Chem 47(4):1209–1223CrossRefGoogle Scholar
  23. Landherr A, Friedl B, Heidemann J (2010) A critical review of centrality measures in social networks. Bus Inf Syst Eng 2(6):371–385CrossRefGoogle Scholar
  24. Leavitt HJ (1951) Some effects of certain communication patterns on group performance. J Abnorm Soc Psychol 46(1):38CrossRefGoogle Scholar
  25. Lindelauf RHA, Hamers HJM, Husslage BGM (2013) Cooperative game theoretic centrality analysis of terrorist networks: the cases of Jemaah Islamiyah and Al Qaeda. Eur J Oper Res 229(1):230–238CrossRefGoogle Scholar
  26. Masuda N, Kawamura Y, Kori H (2009) Analysis of relative influence of nodes in directed networks. Phys Rev E 80(4):046114CrossRefGoogle Scholar
  27. Masuda N, Kori H (2010) Dynamics-based centrality for directed networks. Phys Rev E 82(5):056107CrossRefGoogle Scholar
  28. Meyer CD (2000) Matrix analysis and applied linear algebra. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  29. Mohar B (1991) The Laplacian spectrum of graphs. In: Alavi Y, Chartrand G, Oellermann OR, Schwenk AJ (eds) Graph theory, combinatorics, and applications, vol 2. Wiley, New York, pp 871–898Google Scholar
  30. Monsuur H, Storcken T (2004) Centers in connected undirected graphs: an axiomatic approach. Oper Res 52(1):54–64CrossRefGoogle Scholar
  31. Neumann C (1877) Untersuchungen über das logarithmische und Newton’sche Potential. B. G. Teubner, LeipzigGoogle Scholar
  32. Nieminen J (1974) On the centrality in a graph. Scand J Psychol 15(1):332–336CrossRefGoogle Scholar
  33. Ranjan G, Zhang Z-L (2013) Geometry of complex networks and topological centrality. Phys A Stat Mech Appl 392(17):3833–3845CrossRefGoogle Scholar
  34. Rubinstein A (1980) Ranking the participants in a tournament. SIAM J Appl Math 38(1):108–111CrossRefGoogle Scholar
  35. Sabidussi G (1966) The centrality index of a graph. Psychometrika 31(4):581–603CrossRefGoogle Scholar
  36. Seeley JR (1949) The net of reciprocal influence: a problem in treating sociometric data. Can J Psychol 3(4):234–240CrossRefGoogle Scholar
  37. Wasserman S, Faust K (1994) Social network analysis: methods and applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Operations Research and Actuarial SciencesCorvinus University of BudapestBudapestHungary
  2. 2.MTA-BCE “Lendület” Strategic Interactions Research GroupBudapestHungary

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