Research ArticleAPPLIED MATHEMATICS

Testing for voter rigging in small polling stations

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Science Advances  30 Jun 2017:
Vol. 3, no. 6, e1602363
DOI: 10.1126/sciadv.1602363
  • Fig. 1 Comparison of election fingerprints and their standardized variant for six different elections.

    The SEFs (bottom row) are compared to the election fingerprints (top row) as introduced by Klimek et al. (27) for a selected set of elections, namely (from left to right), Venezuela 1998 and 2013, Russia 2011, Austria 2008, Canada 2011, and Spain 2008. The bimodality in the fingerprint of Canada and the “smearing out” in Austria that can be associated with heterogeneous vote mobilization disappear in the SEF.

  • Fig. 2 Symmetry of the SEFs.

    The SEFs of (A) Venezuela 1998 and (B) 2013, (C) Russia 2011, (D) Russia 2012, (E) Canada 2011, and (F) Spain 2011 are shown using contour representations for the densities of data points. The SEFs appear to be axially symmetric. They are close to the elliptical symmetry.

  • Fig. 3 SEFs for small and large electoral units.

    Comparison of the SEFs of large (blue) and small (red) electoral units for (A) Venezuela 1998 and (B) 2013, (C) Russia 2011, (D) Russia 2012, (E) Canada 2011, and (F) Spain 2011. The centers of the distributions of small and large centers coincide for Venezuela 1998, Canada 2011, and Spain 2011. However, there is a clear discrepancy in the SEFs of small and large centers in Venezuela 2013 and Russia 2011 and 2012. The rescaled turnout and the votes for the winner are substantially larger in small centers for these elections. This is clear evidence that the election outcomes in these small centers show systematic distortions.

  • Fig. 4 Results of the test for voter rigging.

    (A) Results of the statistical test for voter rigging, quantified by the effect size (p), are shown for 21 different elections. The gray region contains the elections with no significant differences between small and large electoral units. These elections are also shown as dash-dotted lines. Solid lines show elections that are compatible with the voter-rigging-in-small-centers hypothesis. These include the elections in Russia, Uganda, and Venezuela from 2006 to 2013, wherein the strongest effects are observed. (B) We show different visualizations of the atypical results observed from Russia, Uganda, and Venezuela. In these visualizations, the electoral units are sorted in a descending way according to their number of electors, and the percentage of votes for the winner is computed using only units up to the given rank on the x axis (in logarithmic scale). For Venezuela 2013, it is only the addition of small units that pushes the results to determine the winner.

  • Fig. 5 Schematic overview of how to compute the Z scores.

    In this illustration, we show electoral units (circles) with different electorate sizes (indicated by the size of the circles). The colors of the units correspond to different electoral neighborhoods. The neighborhood of i is defined as all other units that lie in the same electoral neighborhood (blue lines). To correct for geographic heterogeneities in the data, the vote and turnout percentages of each unit i are rescaled by their average value and SD in the electoral neighborhood of i.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/6/e1602363/DC1

    Data

    Outliers removal

    Visualization of the SEFs

    Modified Thompson tau test

    Discrepancy measures between SEFs of small and large units

    table S1. List of the 21 elections under study with the numbers of electoral units that fulfill the inclusion criteria for our analysis, N.

  • Supplementary Materials

    This PDF file includes:

    • Data
    • Outliers removal
    • Visualization of the SEFs
    • Modified Thompson tau test
    • Discrepancy measures between SEFs of small and large units
    • table S1. List of the 21 elections under study with the numbers of electoral units that fulfill the inclusion criteria for our analysis, N.

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