ABSTRACT
This paper presents several measures of fairness and inequality based on the degree distribution in networks, as alternatives to the well-established power-law exponent. Networks such as social networks, communication networks and the World Wide Web itself are often characterized by their unequal distribution of edges: Few nodes are attached to many edges, while many nodes are attached to only few edges. The inequality of such network structures is typically measured using the power-law exponent, stating that the number of nodes with a given degree is proportional to that degree taken to a certain exponent. However, this approach has several weaknesses, such as its narrow applicability and expensive computational complexity. Beyond the fact that power laws are by far not a universal phenomenon on the Web, the power-law exponent has the surprising property of being negatively correlated with the usual notion of inequality, making it unintuitive as a fairness measure. As alternatives, we propose several measures based on the Lorenz curve, which is used in economics but rarely in networks study, and on the information-theoretical concept of entropy. We show in experiments on a large collection of online networks that these measures do not suffer under the drawbacks of the power-law exponent.
- Albert, R., and Barabási, A.-L. Statistical mechanics of complex networks. Reviews of Modern Physics 74, 1 (2002), 47--97. Google ScholarDigital Library
- Barabási, A.-L., and Albert, R. Emergence of scaling in random networks. Science 286, 5439 (1999), 509--512.Google ScholarCross Ref
- Bi, Z., Faloutsos, C., and Korn, F. The 'DGX' distribution for mining massive, skewed data. In Proc. Int. Conf. on Knowledge Discovery and Data Mining (2001), 17--26. Google ScholarDigital Library
- Bollacker, K., Lawrence, S., and Giles, C. L. CiteSeer: An autonomous Web agent for automatic retrieval and identification of interesting publications. In Proc. Int. Conf. on Autonomous Agents (1998), 116--123. Google ScholarDigital Library
- Bollobás, B. Modern Graph Theory. Springer, 1998.Google ScholarCross Ref
- Celma, Ò. Music recommendation datasets for research. http://www.dtic.upf.edu/~ocelma/MusicRecommendationDataset/lastfm-1K.html, May 2010. Version 1.0.Google Scholar
- Clauset, A., Shalizi, C. R., and Newman, M. E. J. Power-law distributions in empirical data. SIAM Rev. 51, 4 (2009), 661--703. Google ScholarDigital Library
- Kunegis, J., Lommatzsch, A., and Bauckhage, C. The Slashdot Zoo: Mining a social network with negative edges. In Proc. Int. World Wide Web Conf. (2009), 741--750. Google ScholarDigital Library
- Lewis, D. D., Yang, Y., Rose, T. G., and Li, F. RCV1: A new benchmark collection for text categorization research. J. Machine Learning Research 5 (2004), 361--397. Google ScholarDigital Library
- Lima-Mendez, G., and van Helden, J. The powerful law of the power law and other myths in network biology. Molecular BioSystems 5, 12 (2009), 1482--1493.Google ScholarCross Ref
- Manning, C. D., and Schütze, H. Foundations of Statistical Natural Language Processing. MIT Press, 1999. Google ScholarDigital Library
- Massa, P., and Avesani, P. Controversial users demand local trust metrics: an experimental study on epinions.com community. In Proc. American Association for Artificial Intelligence Conf. (2005), 121--126. Google ScholarDigital Library
- Newman, M. E. J. Power laws, Pareto distributions and Zipf's law. Contemporary Phys. 46, 5 (2006), 323--351.Google Scholar
- Pareto, V. Manuale di economia politica con una introduzione alla scienza sociale (Manual of Political Economy). Milano: Societa Editrice Libraria, 1919.Google Scholar
- Viswanath, B., Mislove, A., Cha, M., and Gummadi, K. P. On the evolution of user interaction in Facebook. In Proc. Workshop on Online Social Networks (2009), 37--42. Google ScholarDigital Library
- Wang, B., Tang, H., Guo, C., and Xiu, Z. Entropy optimization of scale-free networks robustness to random failures. Physica A: Statistical Mechanics and its Applications 363, 2 (2006), 591--596.Google ScholarCross Ref
- Wu, J., Tan, Y.-J., Deng, H.-Z., and Zhu, D.-Z. A new measure of heterogeneity of complex networks based on degree sequence. In Unifying Themes in Complex Systems. 2010, 66--73.Google Scholar
Index Terms
- Fairness on the web: alternatives to the power law
Recommendations
Airtime Fairness for IEEE 802.11 Multirate Networks
Under a multi rate network scenario, the IEEE 802.11 DCF MAC fails to provide air-time fairness for all competing stations since the protocol is designed for ensuring max-min throughput fairness and the maximum achievable throughput by any station gets ...
Fairness in multi-hop wireless backhaul networks: a dynamic estimation approach
QShine '08: Proceedings of the 5th International ICST Conference on Heterogeneous Networking for Quality, Reliability, Security and RobustnessIn this work, we consider the problem of fairness for Transit Access Points (TAP) in multi-hop wireless backhaul networks. Existing approaches are not practical due to the requirement for modifications to the MAC layer or queueing operations of TAPs, or ...
Throughput Fairness in Indirect Interconnection Networks
PDCAT '12: Proceedings of the 2012 13th International Conference on Parallel and Distributed Computing, Applications and TechnologiesThe performance of an interconnection network is typically measured by two metrics: average latency and peak network throughput. Average network throughput is usually reported in the belief the network is fair and all source nodes are supposedly able to ...
Comments