Power-law distributions in an online video-sharing system and its long-term dynamics

Abstract We study the data of Japanese video-sharing platform, Niconico, which contains 21 million videos. From our analysis, the rank size distribution of video views is found to exhibit a crossover from a power law with an exponent around $$-0.5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math> for the top $$\approx 10^5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≈</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>5</mml:mn> </mml:msup> </mml:mrow> </mml:math> movies to another power low with exponent around $$-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> for the movies in the following ranks. The probability density function of video views for the bottom $$90\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>90</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> movies is well fitted by log-normal distribution. This implies that, while videos in the top rank regime follow a different dynamics which yields the power law, videos in the middle and low rank regime seem to be evolving according to a random multiplicative process. Furthermore, we observe temporal relaxation process of video views for 3 years. Temporal relaxation process of video views is grouped by the size of the number of video views, and averaged within each size group. Interestingly, the daily video views universally show power-law relaxation in all view size, from the top total view group ( $$10^6-10^7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>7</mml:mn> </mml:msup> </mml:mrow> </mml:math> ) to the bottom group ( $$\approx 10^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≈</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> ). This indicates the existence of memory processes longer than the exponential function, which are universally independent of video size.