Abstract
In some networks nodes belong to predefined groups (e.g., authors belong to institutions). Common network centrality measures do not take this structure into account. Gefura measures are designed as indicators of a node’s brokerage role between such groups. They are defined as variants of betweenness centrality and consider to what extent a node belongs to shortest paths between nodes from different groups. In this article we make the following new contributions to their study: (1) We systematically study unnormalized gefura measures and show that, next to the ‘structural’ normalization that has hitherto been applied, a ‘basic’ normalization procedure is possible. While the former normalizes at the level of groups, the latter normalizes at the level of nodes. (2) Treating undirected networks as equivalent to symmetric directed networks, we expand the definition of gefura measures to the directed case. (3) It is shown how Brandes’ algorithm for betweenness centrality can be adjusted to cover these cases.
Similar content being viewed by others
References
Atkin, R.H., 1972. From cohomology in physics to qconnectivity in social science. Int. J. Man-Mach. Stud., 4(2):139–167. [doi:10.1016/S0020-7373(72)80029-4]
Barrat, A., Barthélemy, M., Pastor-Satorras, R., et al., 2004. The architecture of complex weighted networks. PNAS, 101(11):3747–3752. [doi:10.1073/pnas.0400087101]
Boccaletti, S., Bianconi, G., Criado, R., et al., 2014. The structure and dynamics of multilayer networks. Phys. Rep., 544(1):1–122. [doi:10.1016/j.physrep.2014.07.001]
Brainard, W.C., Tobin, J., 1968. Econometric models: their problems and usefulness. Pitfalls in financial model building. Amer. Econ. Rev., 58(2):99–122.
Brandes, U., 2001. A faster algorithm for betweenness centrality. J. Math. Sociol., 25(2):163–177. [doi:10.1080/0022250X.2001.9990249]
Brandes, U., 2008. On variants of shortest-path betweenness centrality and their generic computation. Soc. Netw., 30(2):136–145. [doi:10.1016/j.socnet.2007.11.001]
Burt, R.S., 2004. Structural holes and good ideas. Amer. J. Sociol., 110(2):349–396. [doi:10.1086/421787]
Chen, L.X., Rousseau, R., 2008. Q-measures for binary divided networks: bridges between German and English institutes in publications of the J. Fluid Mech. Scientometr., 74(1):57–69. [doi:10.1007/s11192-008-0103-6]
Christensen, C., Albert, R., 2007. Using graph concepts to understand the organization of complex systems. Int. J. Bifurc. Chaos, 17(7):2201–2214. [doi:10.1142/S021812740701835X]
Ding, Y., 2011. Scientific collaboration and endorsement: network analysis of coauthorship and citation networks. J. Inform., 5(1):187–203. [doi:10.1016/j.joi.2010.10.008]
Flom, P.L., Friedman, S.R., Strauss, S., et al., 2004. A new measure of linkage between two sub-networks. Connections, 26(1):62–70.
Freeman, L.C., 1977. A set of measures of centrality based on betweenness. Sociometry, 40(1):35–41.
Freeman, L.C., Borgatti, S.P., White, D.R., 1991. Centrality in valued graphs: a measure of betweenness based on network flow. Soc. Netw., 13(2):141–154. [doi:10.1016/0378-8733(91)90017-N]
Gould, R.V., Fernandez, R.M., 1989. Structures of mediation: a formal approach to brokerage in transaction networks. Sociol. Method., 19:89–126. [doi:10.2307/270949]
Guimerà, R., Amaral, L.A.N., 2005. Functional cartography of complex metabolic networks. Nature, 433:895–900. [doi:10.1038/nature03288]
Guimerà, R., Mossa, S., Turtschi, A., et al., 2005. The worldwide air transportation network: anomalous centrality, community structure, and cities’ global roles. PNAS, 102(22):7794–7799. [doi:10.1073/pnas.0407994102]
Guns, R., Liu, Y.X., 2010. Scientometric research in China in context of international collaboration. Proc. 6th Int. Conf. on Scientometrics and University Evaluation, p.112–115.
Guns, R., Rousseau, R., 2009. Gauging the bridging function of nodes in a network: Q-measures for networks with a finite number of subgroups. Proc. 12th ISSI, p.131–142.
Guns, R., Liu, Y.X., Mahbuba, D., 2011. Q-measures and betweenness centrality in a collaboration network: a case study of the field of informetrics. Scientometrics, 87(1):133–147. [doi:10.1007/s11192-010-0332-3]
Liu, Y.X., Guns, R., Rousseau, R., 2013. A binary tree as a basic model for studying hierarchies using Q-measures. SRELS J. Inform. Manag., 50(5):521–528.
Newman, M.E.J., Girvan, M., 2004. Finding and evaluating community structure in networks. Phys. Rev. E, 69: 026113.1–026113.15. [doi:10.1103/PhysRevE.69.026113]
Otte, E., Rousseau, R., 2002. Social network analysis: a powerful strategy, also for the information sciences. J. Inform. Sci., 28(6):441–453. [doi:10.1177/016555150202800601]
Rousseau, R., 2005. Q-measures for binary divided networks: an investigation within the field of informetrics. Proc. Amer. Soc. Inform. Sci. Technol., 42(1):675–696.
Rousseau, R., Zhang, L., 2008. Betweenness centrality and Qmeasures in directed valued networks. Scientometrics, 75(3):575–590. [doi:10.1007/s11192-007-1772-2]
Rousseau, R., Liu, Y.X., Guns, R., 2013. Mathematical properties of Q-measures. J. Inform., 7(3):737–745. [doi:10.1016/j.joi.2013.06.002]
Rousseau, R., Liu, Y.X., Guns, R., 2014. An addendum and correction to “Mathematical properties of Q-measures” (vol. 7, issue 3, pp.737–745). J. Inform., 8(3):486–490. [doi:10.1016/j.joi.2014.01.004]
Rousseau, R., Guns, R., Liu, Y.X., 2015. Gauging the bridging function of nodes in a network: the gefura measure. Proc. 8th Int. Conf. on Scientometrics and University Evaluation, in press.
Sakai, T., 2007. On the reliability of information retrieval metrics based on graded relevance. Inform. Process. Manag., 43(2):531–548. [doi:10.1016/j.ipm.2006.07.020]
Wasserman, S., Faust, K., 1994. Social Network Analysis: Methods and Applications. Cambridge University Press, UK.
Zhang, W.L., Yin, L.C., Pang, J., 2009. The application of Qmeasure to gender study in cooperation network. Sci. Technol. Progr. Pol., 26(15):100–103 (in Chinese).
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 71173154)
ORCID: Raf GUNS, http://orcid.org/0000-0003-3129-0330; Ronald ROUSSEAU, http://orcid.org/0000-0002-3252-2538
Rights and permissions
About this article
Cite this article
Guns, R., Rousseau, R. Unnormalized and normalized forms of gefura measures in directed and undirected networks. Frontiers Inf Technol Electronic Eng 16, 311–320 (2015). https://doi.org/10.1631/FITEE.1400425
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1631/FITEE.1400425
Key words
- Networks subdivided in groups
- Partitions
- Gefura measures
- Q-measures
- Brokerage role
- Directed and undirected networks
- Brandes’ algorithm