Stochastic dynamics and the predictability of big hits in online videos

The competition for the attention of users is a central element of the Internet. Crucial issues are the origin and predictability of big hits, the few items that capture a big portion of the total attention. We address these issues analyzing ${10}^6$ time series of videos' views from YouTube. We find that the average gain of views is linearly proportional to the number of views a video already has, in agreement with usual rich-get-richer mechanisms and Gibrat's law, but this fails to explain the prevalence of big hits. The reason is that the fluctuations around the average views are themselves heavy tailed. Based on these empirical observations, we propose a stochastic differential equation with Lévy noise as a model of the dynamics of videos. We show how this model is substantially better in estimating the probability of an ordinary item becoming a big hit, which is considerably underestimated in the traditional proportional-growth models.

Figure 1

Evolution of videos' views $X_t$ as a function of the time $t$ after publication. After $t=3$ days the distribution of views is already heavy tailed (orange histogram). Videos having initially the same amount of views show very distinct evolutions. This is illustrated here by highlighting two groups of videos with the same number of views at $t=3$ (blue $X_{t=3}=50$ and green $X_{t=3}=100$). Each line (at the bottom and in the right plot) corresponds to the trajectory of one video. On the back, the histograms of the two groups of videos at $t=20$ days are shown.

Figure 2

Average and fluctuations in the growth of YouTube videos. (a) Mean $\langle \mathrm{\Delta }{X}_t\rangle$ and (b) standard deviation $\sigma (\mathrm{\Delta }{X}_t$) for videos with $X_t$ views $t=3$ days after publication. Both $\langle ...\rangle$ and $\sigma$ are computed in windows centered at $X_t$ and containing $N$ items (see legend). (c) Complementary cumulative density function $F\left(\mathrm{\Delta }{X}_t\right|X_t)$ for $X_t\in [499,513]$ and $t=3$. Data (blue circles) is compared with fits of three distributions (S: Lévy-stable, LN: log-normal, CEV: constant elasticity of variance) and confirms the existence of heavy tails, which are best described by the Lévy distribution.

Figure 3

Agreement of the model with respect to data. Main panel: complementary cumulative distribution of the views rescaled by the fitted parameters for $t=3$. The rescaled histograms $\mathbb{P}\left(\mathrm{\Delta }{X}_3\right|X_3)$ are plotted as points, where each color corresponds to a different value of $X_3$; the black line is the Lévy-stable distribution with location 0 and scale 1. Inset: BIC difference with respect to the S model [41].

Figure 4

Evolution of the parameters for the S model, Eq. (2), in the first 30 days after the release.

Figure 5

Probability of videos becoming a big hit. Performance of the models evolved in time with respect to data; the selected videos had 100 views one day after their publication. Main panel: amount of videos that exceed a threshold $x$ at $t=6$. In the top axis, the quantiles $q$ are indicated. Inset: amount of videos that enter into the $5\%$ most viewed. Shaded areas: 95% confidence intervals, by bootstrapping.