Abstract
The centrality of a node within a network, however it is measured, is a vital proxy for the importance or influence of that node, and the differences in node centrality generate hierarchies and inequalities. If the network is evolving in time, the influence of each node changes in time as well, and the corresponding hierarchies are modified accordingly. However, there is still a lack of systematic study into the ways in which the centrality of a node evolves when a graph changes. In this paper we introduce a taxonomy of metrics of equality and hierarchical mobility in networks that evolve in time. We propose an indicator of equality based on the classical Gini Coefficient from economics, and we quantify the hierarchical mobility of nodes, that is, how and to what extent the centrality of a node and its neighbourhood change over time. These measures are applied to a corpus of thirty time evolving network data sets from different domains. We show that the proposed taxonomy measures can discriminate between networks from different fields. We also investigate correlations between different taxonomy measures, and demonstrate that some of them have consistently strong correlations (or anti-correlations) across the entire corpus. The mobility and equality measures developed here constitute a useful toolbox for investigating the nature of network evolution, and also for discriminating between different artificial models hypothesised to explain that evolution.
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Notes
- 1.
We introduce the term hierarchical mobility to avoid confusion with the word “mobility” alone, commonly used in spatial networks as a measure of nodes’ ability to move in geographic space.
- 2.
The code and data set collection information is stored here https://github.com/matthewrussellbarnes/mobility_taxonomy.
- 3.
Technically this should be called “anti”-mobility as a larger correlation coefficient refers to fewer changes in hierarchical position.
- 4.
This is not quite correct as the assortativity at \(t_2\) would be measured on the neighbourhood set at \(t_2\) not the neighbourhood set at \(t_1\).
- 5.
See footnote 2.
- 6.
Some networks were originally above 1 million edges but here are truncated to keep computational time reasonable.
References
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999). https://doi.org/10.1126/science.286.5439.509
Breen, R.: Social Mobility in Europe. OUP Oxford (2004)
Clegg, R.G., et al.: Measuring the likelihood of models for network evolution. In: Proceedings of INFOCOM’09, pp. 272–277 (2009)
Corman, S.R., Kuhn, T., Mcphee, R.D.: Studying complex discursive systems. Hum. Commun. Res. (2002). https://doi.org/10.1111/j.1468-2958.2002.tb00802.x
Dimaggio, P., Garip, F.: Network effects and social inequality. Annu. Rev. Sociol. 38, 93–118 (2012). https://doi.org/10.1146/annurev.soc.012809.102545
Erikson, R., Goldthorpe, J.H.: The Constant Flux: A Study of Class Mobility in Industrial Societies. Oxford University Press (1992)
Fire, M., Guestrin, C.: The rise and fall of network stars. Inf. Process. Manag. 57(2) (2020). https://doi.org/10.1016/j.ipm.2019.05.002
Fortunato, S., et al.: Scale-free network growth by ranking. Phys. Rev. Lett. 96(21), 1–4 (2006). https://doi.org/10.1103/PhysRevLett.96.218701
Fowler, J.H., et al.: Network analysis and the law. Polit. Anal. 15(3), 324–346 (2007). https://doi.org/10.1093/pan/mpm011
Génois, M., et al.: Data on face-to-face contacts in an office building suggest a low-cost vaccination strategy based on community linkers. Netw. Sci. 3(3), 326–347 (2015). https://doi.org/10.1017/nws.2015.10
Gini, C.: Variabilità e mutabilità. Memorie di metodologica statistica (1912)
Granovetter, M.S.: The strength of weak ties. Am. J. Sociol. 78(6), 1360–1380 (1973)
Hall, B.H., et al: The NBER Patent Citation Data File: Lessons, Insights and Methodological Tools. NBER Cambridge, Mass (2001). https://doi.org/10.3386/w8498
Holanda, A.J., et al: Character networks and book genre classification. IJMPC 30(8) (2019). https://doi.org/10.1142/S012918311950058X
Iñiguez, G., Pineda, C., Gershenson, C., Barabási, A.L.: Universal dynamics of ranking. Nat. Commun. (2021). http://arxiv.org/abs/2104.13439
Isella, L., et al.: What’s in a crowd? analysis of face-to-face behavioral networks. JTB 271(1), 166–180 (2011). https://doi.org/10.1016/j.jtbi.2010.11.033
Kumar, S., et al: Community interaction and conflict on the web. In: Proceedings of the 2018 World Wide Web Conference, pp. 933–943 (2018). https://doi.org/10.1145/3178876.3186141
Lim, E.P., et al.: Detecting product review spammers using rating behaviors. In: Proceedings of the 19th ACM International Conference on Information and Knowledge Management, pp. 939–948 (2010)
Lorenz, M.O.: Methods of measuring the concentration of wealth. Publ. Am. Stat. Assoc. 9(70), 209–219 (1905)
Mayer, S.E., Lopoo, L.M.: Has the intergenerational transmission of economic status changed? JHR 40(1), 169–185 (2005). https://doi.org/10.3368/jhr.xl.1.169
Nsour, F., Sayama, H.: Hot-get-richer network growth model. In: International Conference on Complex Networks and Their Applications, pp. 532–543. Springer, Berlin (2020)
Panzarasa, P., et al.: Patterns and dynamics of users’ behavior and interaction: network analysis of an online community. J. Am. Soc. Inf. Sci. Technol. 60(5), 911–932 (2009)
Paranjape, A., et al: Motifs in temporal networks. In: Proceedings of the tenth ACM International Conference on Web Search and Data Mining, pp. 601–610 (2017). https://doi.org/10.1145/3018661.3018731
Park, D., et al: Novelty and influence of creative works, and quantifying patterns of advances based on probabilistic references networks. EPJ Data Sci. 9(1) (2020). https://doi.org/10.1140/epjds/s13688-019-0214-8
Redner, S.: Citation statistics from 110 years of physical review. Phys. Today 58(6), 49–54 (2005). https://doi.org/10.1063/1.1996475
Solon, G.: Intergenerational Income Mobility in the United States. The American Economic Review, pp. 393–408 (1992)
Stehlé, J., et al: High-resolution measurements of face-to-face contact patterns in a primary school. PLoS One 6(8) (2011). https://doi.org/10.1371/journal.pone.0023176
Szreter, S.R.S.: The genesis of the registrar-general’s social classification of occupations. Br. J. Sociol. 35(4), 522–546 (1984)
Taylor, D., et al.: Eigenvector-based centrality measures for temporal networks. MMS 15(1), 537–574 (2017). https://doi.org/10.1137/16M1066142
Valverde, S., Sole, R.V.: Punctuated equilibrium in the large-scale evolution of programming languages. JRSI 12(107) (2015). https://doi.org/10.1098/rsif.2015.0249
Vanhems, P., et al: Estimating potential infection transmission routes in hospital wards using wearable proximity sensors. PloS One 8(9) (2013). https://doi.org/10.1371/journal.pone.0073970
Viswanath, B., et al.: On the evolution of user interaction in facebook. In: Proceedings of the 2nd ACM Workshop on Online Social Networks, pp. 37–42 (2009)
Wachs, J., et al.: Corruption risk in contracting markets. IJDSA 12(1), 45–60 (2021). https://doi.org/10.1007/s41060-019-00204-1
Zhou, B., Lu, X., Holme, P.: Universal evolution patterns of degree assortativity in social networks. Soc. Netw. 63, 47–55 (2020). https://doi.org/10.1016/j.socnet.2020.04.004
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The authors wish to acknowledge the support of Moogsoft Ltd. for funding this research.
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Barnes, M.R., Nicosia, V., Clegg, R.G. (2022). Measuring Equality and Hierarchical Mobility on Abstract Complex Networks. In: Pacheco, D., Teixeira, A.S., Barbosa, H., Menezes, R., Mangioni, G. (eds) Complex Networks XIII. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-031-17658-6_2
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