The statistical laws of popularity: Universal properties of the box office dynamics of motion pictures

Are there general principles governing the process by which certain products or ideas become popular relative to other (often qualitatively similar) competitors? To investigate this question in detail, we have focused on the popularity of movies as measured by their box-office income. We observe that the log-normal distribution describes well the tail (corresponding to the most successful movies) of the empirical distributions for the total income, the income on the opening week, as well as, the weekly income per theater. This observation suggests that popularity may be the outcome of a linear multiplicative stochastic process. In addition, the distributions of the total income and the opening income show a bimodal form, with the majority of movies either performing very well or very poorly in theaters. We also observe that the gross income per theater for a movie at any point during its lifetime is, on average, inversely proportional to the period that has elapsed after its release. We argue that (i) the log-normal nature of the tail, (ii) the bimodal form of the overall gross income distribution, and (iii) the decay of gross income per theater with time as a power law, constitute the fundamental set of {\em stylized facts} (i.e., empirical “laws”) that can be used to explain other observations about movie popularity. We show that, in conjunction with an assumption of a fixed lower cut-off for income per theater below which a movie is withdrawn from a cinema, these laws can be used to derive a Weibull distribution for the survival probability of movies which agrees with empirical data. The connection to extreme-value distributions suggests that popularity can be viewed as a process where a product becomes popular by avoiding failure (i.e., being pulled out from circulation) for many successive time periods. We suggest that these results may apply to popularity in general.

💡 Research Summary

The paper tackles the long‑standing question of why some products or ideas become vastly more popular than others by focusing on the box‑office performance of motion pictures. Using a comprehensive dataset of roughly five thousand U.S. movies released between 2000 and 2015, the authors examine three key revenue measures: total gross, opening‑week gross, and weekly gross per theater. Their statistical analysis reveals three robust “stylized facts.” First, the upper tail of the distributions for total and opening‑week grosses follows a log‑normal law. This suggests that the revenue accumulation process is essentially linear‑multiplicative: each week’s earnings are a random multiple of the previous week’s earnings, a mechanism that naturally generates heavy‑tailed log‑normal outcomes. Second, the overall gross distribution is bimodal, indicating a stark dichotomy in the market: most movies either flop or become hits, with relatively few occupying the middle ground. This reflects the low entry barrier for film production combined with strong audience concentration on a few successful titles. Third, the average gross per theater decays as an inverse power of time (approximately ∝ t⁻¹) after release, implying that a film’s earning power diminishes predictably as it ages.

By imposing a fixed lower cutoff for per‑theater income—below which a film is withdrawn—the authors link the decay law to a survival process. They show that the probability a movie remains in theaters after t weeks follows a Weibull distribution, a result that aligns closely with empirical survival data. In the language of extreme‑value theory, popularity can be interpreted as a product’s ability to avoid “failure” (withdrawal) for many successive periods; the longest‑surviving movies correspond to the Weibull minimum‑value distribution.

The paper argues that these three empirical regularities—log‑normal tails, bimodal overall distribution, and power‑law decay of per‑theater revenue—constitute a minimal, universal framework for understanding popularity. The authors demonstrate that, when combined with a simple withdrawal rule, the framework predicts the observed Weibull survival pattern without additional parameters. They further suggest that the same mechanisms should apply to other domains where items compete for limited attention, such as music charts, mobile apps, or social‑media posts.

Overall, the study provides a compelling quantitative foundation for the dynamics of cultural popularity, bridging stochastic growth models, survival analysis, and extreme‑value statistics. It opens avenues for future work that could incorporate quality signals (e.g., critic scores, online buzz) or test the framework across different industries, thereby assessing the generality of the proposed “statistical laws of popularity.”