The Inequality of Success for the Rankings.

Abstract

<p>The success of song <i>i</i> is defined as the market share <i>m</i>_<i>i</i> for that song, i.e., </p><p></p><p></p><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo>=</mo><p><mi>D</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo>/</mo><p><mo>∑</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mi>D</mi></p><p><mi>j</mi><mo>,</mo><mi>k</mi></p><p></p><p></p><p></p> for a given world <i>k</i>. The success inequality is defined by the Gini coefficient <p></p><p><mi>G</mi><mo>=</mo></p><p><mo>∑</mo></p><p><mi>i</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mo>∑</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><mo stretchy="false">∣</mo><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo>−</mo><p><mi>m</mi></p><p><mi>j</mi><mo>,</mo><mi>k</mi></p><p></p><mo stretchy="false">∣</mo><mo>/</mo><mn>2</mn><mi>n</mi><p><mo>∑</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mi>m</mi></p><p><mi>j</mi><mo>,</mo><mi>k</mi></p><p></p><p></p><p></p>, which represents the average difference of market share for all songs. The figure depicts a boxplot with whiskers from minimum to maximum. The results for the first setting are in black, while the results for the second setting are in red and dashed line.<p></p