Measuring centrality by a generalization of degree
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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.
KeywordsNetwork Centrality measure Degree Axiomatic approach
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.
- 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
- Chebotarev P (1989) Generalization of the row sum method for incomplete paired comparisons. Autom Remote Control 50(8):1103–1113Google Scholar
- Chebotarev P, Shamis E (1997) The matrix-forest theorem and measuring relations in small social groups. Autom Remote Control 58(9):1505–1514Google Scholar
- Dequiedt V, Zenou Y (2015) Local and consistent centrality measures in networks. https://www.gate.cnrs.fr/IMG/pdf/Dequiedt2015.pdf
- Garg M (2009) Axiomatic foundations of centrality in networks. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1372441
- 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
- Jackson MO (2010) Social and economic networks. Princeton University Press, PrincetonGoogle Scholar
- Kitti M (2012) Axioms for centrality scoring with principal eigenvectors. http://www.ace-economics.fi/kuvat/dp79.pdf
- 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
- Neumann C (1877) Untersuchungen über das logarithmische und Newton’sche Potential. B. G. Teubner, LeipzigGoogle Scholar