Quality & Quantity

, Volume 52, Issue 3, pp 1069–1079 | Cite as

A multiplicative process for generating the rank-order distribution of UK election results

  • Trevor Fenner
  • Mark Levene
  • George Loizou


Human dynamics and sociophysics suggest statistical models that may explain and provide us with a better understanding of social phenomena. Here we propose a generative multiplicative decrease model that gives rise to a rank-order distribution and allows us to analyse the results of the last three UK parliamentary elections. We provide empirical evidence that the additive Weibull distribution, which can be generated from our model, is a close fit to the electoral data, offering a novel interpretation of the recent election results.


Election results Generative model Multiplicative process Rank-order distribution Additive Weibull distribution 



We would like to thank Muawya Eldaw, who preprocessed the election data sets.


  1. Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)CrossRefGoogle Scholar
  2. Alvarez, R.M., Nagler, J.: When politics and models collide: estimating models of multiparty elections. Am. J. Polit. Sci. 42, 55–96 (1998)CrossRefGoogle Scholar
  3. Barabási, A.-L.: The architecture of complexity: from network structure to human dynamics. IEEE Control Syst. Mag. 27, 33–42 (2007)CrossRefGoogle Scholar
  4. Brakman, S., Garretsen, H., Van Marrewijk, C., Van den Burg, M.: The return of Zipf: towards a further understanding of the rank-size distribution. J. Reg. Sci. 39, 183–213 (1999)CrossRefGoogle Scholar
  5. Börner, S., Sanyal, S., Vespignani, A.: Network science. Annu. Rev. Inf. Sci. Technol. (ARIST) 41, 537–607 (2007)CrossRefGoogle Scholar
  6. Curtice, J., Firth, D.: Exit polling in a cold climate: the BBCITV experience in Britain in 2005. J. R. Stat. Soc. Ser. A (Stat. Soc.) 171, 509–539 (2008)CrossRefGoogle Scholar
  7. Curtice, J., Fisher, S.D., Kuha, J.: Confounding the commentators: how the 2010 exit poll got it (more or less) right. J. Elections Public Opin. Parties 21, 211–235 (2011)CrossRefGoogle Scholar
  8. Charlesworth, B.: Evolution in Age-Structured Populations. Cambridge Studies in Mathematical Biology, 2nd edn. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  9. Chatterjee, A., Mitrović, M., Fortunato, S.: Universality in voting behavior: an empirical analysis. Nat. Sci. Rep. 3, 1049 (2013)CrossRefGoogle Scholar
  10. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley Series in Telecommunications. Wiley, Chichester (1991)CrossRefGoogle Scholar
  11. Chen, C.-C., Tseng, C.-Y., Telesca, L., Chi, S.-C., Sun, L.-C.: Collective Weibull behavior of social atoms: application of the rank-ordering statistics to historical extreme events. Europhys. Lett. 97, 48010-1–48010-6 (2012)Google Scholar
  12. Da Paz, R.F., Ehlers, R.S., Bazán, J.L.: A Weibull mixture model for the votes of a Brazilian political party, chapter 19. In: Polpo, A., Louzada, F., Rifo, L.L.R., Stern, J.M., Lauretto, M. (eds.) Interdisciplinary Bayesian Statistics, EBEB 2014, Volume 118 of Springer Proceedings in Mathematics & Statistics, pp. 229–241. Springer, Cham (2015)Google Scholar
  13. Denver, D.: The results: how Britian voted. In: Geddes, A., Tonge, J. (eds.) British Votes 2015, pp. 5–24. Oxford University Press, Oxford (2015)Google Scholar
  14. Eeckhout, J.: Gibrat’s law for (all) cities. Am. Econ. Rev. 94, 1429–1451 (2004)CrossRefGoogle Scholar
  15. Endres, D., Schindelin, J.: A new metric for probability distributions. IEEE Trans. Inf. Theory 49, 1858–1860 (2003)CrossRefGoogle Scholar
  16. Fenner, T., Levene, M., Loizou, G.: A model for collaboration networks giving rise to a power-law distribution with an exponential cutoff. Soc. Netw. 29, 70–80 (2007)CrossRefGoogle Scholar
  17. Fenner, T., Levene, M., Loizou, G.: A discrete evolutionary model for chess players ratings. IEEE Trans. Comput. Intell. AI Games 4, 84–93 (2012)CrossRefGoogle Scholar
  18. Fenner, T., Levene, M., Loizou, G.: A bi-logistic growth model for conference registration with an early bird deadline. Cent. Eur. J. Phys. 11, 904–909 (2013)Google Scholar
  19. Fenner, T., Levene, M., Loizou, G.: A stochastic evolutionary model for capturing human dynamics. J. Stat. Mech. Theory Exp. 2015, P08015 (2015)CrossRefGoogle Scholar
  20. Galam, S.: Sociophysics: a review of Galam models. J. Mod. Phys. C 19, 409–440 (2008)CrossRefGoogle Scholar
  21. Horowitz, D.L.: Electoral systems: a primer for decision makers. J. Democr. 14, 115–127 (2003)CrossRefGoogle Scholar
  22. Hedström, P., Swedberg, R.: Social mechanisms: an introductory essay. In: Hedström, P., Swedberg, R. (eds.) Social Mechanisms: An Analytical Approach to Social Theory, pp. 1–31. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  23. Johnson, N.L., Kotz, S., Balkrishnan, N.: Continuous Univariate Distributions. Wiley Series in Probability and Mathematical Statistics, vol. 1, 2nd edn. Wiley, New York (1994)Google Scholar
  24. Kleinbaum, D.G., Klein, M.: Survival Analysis: A Self-Learning Text, 3rd edn. Springer, LLC, New York (2012)CrossRefGoogle Scholar
  25. Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc. 53, 457–481 (1958)CrossRefGoogle Scholar
  26. Laherrère, J., Sornette, D.: Stretched exponential distributions in nature and economy: fat tails with characteristic scales. Eur. Phys. J. B 2, 525–539 (1998)CrossRefGoogle Scholar
  27. Lai, C.D.: Generalized Weibull Distributions. Springer Briefs in Statistics. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  28. Lawless, J.F.: Statistical Models and Methods for Lifetime Data. Wiley Series in Probability and Statistics, 2nd edn. Wiley, New York (2003)Google Scholar
  29. Lax, P.D.: Hyperbolic Partial Differential Equations. Courant Lecture Notes. American Mathematical Society, Providence (2006)Google Scholar
  30. Li, J., Brauer, F.: Continuous-time age-structured models in population dynamics and epidemiology, chapter 9. In: Brauer, F., Van den Driessche, P., Wu, J. (eds.) Mathematical Epidemiology. Lecture Notes in Mathematics, Mathematical Biosciences Subseries, pp. 205–227. Springer, Berlin (2008)Google Scholar
  31. Lemonte, A.J., Cordeiro, G.M., Ortega, E.M.M.: On the additive Weibull distribution. Commun. Stat. Theory Methods 43, 2066–2080 (2014)CrossRefGoogle Scholar
  32. Limpert, E., Stahel, W.A., Abbt, M.: Log-normal distributions across the sciences: keys and clues. BioScience 51, 341–352 (2001)CrossRefGoogle Scholar
  33. Manning, C.D., Schütze, H. (eds.): Foundations of Statistical Natural Language Processing. MIT Press, Cambridge (1999)Google Scholar
  34. Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Math. 1, 226–251 (2004)CrossRefGoogle Scholar
  35. Motulsky, H.: Intuitive Biostatistics. Oxford University Press, Oxford (1995)Google Scholar
  36. Nadarajah, S.: Bathtub-shaped failure rate functions. Qual. Quant. 43, 855–863 (2009)CrossRefGoogle Scholar
  37. O’Connor, P.D.T., Kleyner, A.: Practical Reliability Engineering. Wiley Series in Telecommunications, 5th edn. Wiley, Chichester (2012)Google Scholar
  38. Pan, R.K., Sinha, S.: The statistical laws of popularity: universal properties of the box-office dynamics of motion pictures. New J. Phys. 12, 115004 (2010)CrossRefGoogle Scholar
  39. Pilant, M., Rundell, W.: Determining a coefficient in a first-order hyperbolic equation. SIAM J. Appl. Math. 51, 494–506 (1991)CrossRefGoogle Scholar
  40. Redner, S.: Random multiplicative processes: an elementary tutorial. Am. J. Phys. 3, 267–273 (1990)CrossRefGoogle Scholar
  41. Rinne, H.: The Weibull Distribution: A Handbook. CRC Press, Boca Raton (2009)Google Scholar
  42. Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, New York (1996)Google Scholar
  43. Schelling, T.C.: Social mechanisms and social dynamics. In: Hedström, P., Swedberg, R. (eds.) Social Mechanisms: An Analytical Approach to Social Theory, pp. 32–44. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  44. Sen, P., Chakrabarti, B.K.: Sociophysics: An Introduction. Oxford University Press, Oxford (2014)Google Scholar
  45. Simon, H.A.: On a class of skew distribution functions. Biometrika 42, 425–440 (1955)CrossRefGoogle Scholar
  46. Sornette, D., Knopoff, L., Kagan, Y.Y., Vanneste, C.: Rank-ordering statistics of extreme events: application to the distribution of large earthquakes. J. Geophys. Res. 101, 13-883–13-893 (1996)CrossRefGoogle Scholar
  47. Stein, W.E., Dattero, R.: A new discrete Weibull distribution. IEEE Trans. Reliab. 33, 196–197 (1984)CrossRefGoogle Scholar
  48. Stauffer, D.: Statistical physics for humanists: a tutorial, chapter 18. In: Burguete, M., Lam, L. (eds.) All About Science, History, Philosophy, Sociology & Communications, Science Matters, pp. 383–406. World Scientific, Singapore (2014)Google Scholar
  49. Tsoularis, A., Wallace, J.: Analysis of logistic growth models. Math. Biosci. 179, 21–55 (2002)CrossRefGoogle Scholar
  50. Xie, M., Lai, C.D.: Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Saf. 52, 87–93 (1995)CrossRefGoogle Scholar
  51. Zanette, D.H.: Multiplicative processes and city sizes. In: Albeverio, S., Andrey, D., Giordano, P., Vancheri, A. (eds.) The Dynamics of Complex Urban Systems An Interdisciplinary Approach, pp. 457–472. Physica-Verlag, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Information SystemsBirkbeck, University of LondonLondonUK

Personalised recommendations