Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T01:16:15.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Commentary: Measuring the shape of degree distributions

Published online by Cambridge University Press:  28 August 2013

JENNIFER M. BADHAM
JENNIFER M. BADHAM*
Affiliation:
School of Engineering and Information Technology, Australian Defence Force Academy, Northcott Drive, Canberra ACT 2600, Australia (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Degree distribution is a fundamental property of networks. While mean degree provides a standard measure of scale, there are several commonly used shape measures. Widespread use of a single shape measure would enable comparisons between networks and facilitate investigations about the relationship between degree distribution properties and other network features. This paper describes five candidate measures of heterogeneity and recommends the Gini coefficient. It has theoretical advantages over many of the previously proposed measures, is meaningful for the broad range of distribution shapes seen in different types of networks, and has several accessible interpretations. While this paper focuses on degree, the distribution of other node-based network properties could also be described with Gini coefficients.

Keywords

degree distributionheterogeneitycentralizationGini coefficient
Type
Article Commentary
Information
Network Science , Volume 1 , Issue 2 , August 2013 , pp. 213 - 225
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albert, R., Jeong, H. & Barabási, A.-L. (1999). Diameter of the world wide web. Nature (London), 401 (6749), 130–131.Google Scholar
Allison, P. D. (1978). Measures of inequality. American Sociological Review, 43 (6), 865–880.Google Scholar
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512.Google Scholar
Brinkmeier, M., & Schank, T. (2005). Network statistics. In Brandes, U. & Erlebach, T. (Eds.), Network Analysis (pp. 293–317). Springer-Verlag.Google Scholar
Brown, M. C. (1994). Using Gini style indices to evaluate the spatial patterns of health practitioners: Theoretical considerations and an application based on Alberta data. Social Science and Medicine, 38 (9), 1243–1256.CrossRefGoogle Scholar
Butts, C. T. (2006). Exact bounds for degree centralization. Social Networks, 28, 283–296.Google Scholar
Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51 (4), 661–703.Google Scholar
Coleman, J. S. (1964). Introduction to Mathematical Sociology. Free Press (MacMillan).Google Scholar
Cowell, F. A. (2000). Measurement of inequality. In Atkinson, A. B., & Bourguignon, F. (Eds.), Handbook of Income Distribution (pp. 87–166). Elsevier.Google Scholar
Dalton, H. (1920). The measurement of the inequality of incomes. The Economic Journal, 30 (119), 348–361.CrossRefGoogle Scholar
Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28, 365–382.Google Scholar
Dorfman, R. (1979). A formula for the Gini coefficient. The Review of Economics and Statistics, 61 (1), 146–149.CrossRefGoogle Scholar
Erdös, P., & Rényi, A. (1960). On the evolution of random graphs. Publications of the Institute of Mathematics, Hungarian Academy of Science, 5, 17–60.Google Scholar
Freeman, L. C. (1978). Centrality in social networks: Conceptual clarification. Social Networks, 1, 215–239.Google Scholar
Gini, C. (1912). Variabilità e mutabilità. Studi Economico-Giuridici dell'Università di Cagliari, 3, 1–158.Google Scholar
Hakimi, S. L. (1962). On realizability of a set of integers as degrees of the vertices of a linear graph. Journal of the Society for Industrial and Applied Mathematics, 10 (3), 496–506.Google Scholar
Hirschman, A. O. (1964). The paternity of an index. American Economic Review, 54 (5), 761–762.Google Scholar
Hu, H. B., & Wang, X. F. (2008). Unified index to quantifying heterogeneity of complex networks. Physica A: Statistical Mechanics and Its Applications, 387 (14), 3769–3780.Google Scholar
Jeong, H., Mason, S. P., Barabási, A.-L., & Oltvai, Z. N. (2001). Lethality and centrality in protein networks. Nature (London), 411 (6833), 41–42.Google Scholar
Lopes, G. R., da Silva, R., & de Oliveira, J. P. M. (2011). Applying Gini coefficient to quantify scientific collaboration in researchers network. Proceedings of the International Conference on Web Intelligence, Mining and Semantics, WIMS '11 (pp. 68:1–6). New York: ACM.Google Scholar
Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9 (70), 209–219.CrossRefGoogle Scholar
Massey, F. J. Jr (1951). The Kolmogorov-Smirnov test for goodness of fit. Journal of the American Statistical Association, 46 (253), 68–78.Google Scholar
Newman, M. E. J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences of the United States of America, 98, 404–409.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2002). Assortative mixing in networks. Physical Review Letters, 89, 208701.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45 (2), 167–256.CrossRefGoogle Scholar
Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46 (5), 323–351.Google Scholar
Rapoport, A., & Horvath, W. J. (1961). A study of a large sociogram. Behavioral Science, 6 (4), 279–291.Google Scholar
Shannon, C. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27, 379–423.Google Scholar
Snijders, T. A. B. (1981). The degree variance: An index of graph heterogeneity. Social Networks, 3 (3), 163–174.Google Scholar