Dynamics of movie competition and popularity spreading in recommender systems

We introduce a simple model to study movie competition in recommender systems. Movies of heterogeneous quality compete against each other through viewers’ reviews and generate interesting dynamics at the box office. By assuming mean-field interactions between the competing movies, we show that the runaway effect of popularity spreading is triggered by defeating the average review score, leading to box-office hits: Popularity rises and peaks before fade-out. The average review score thus characterizes the critical movie quality necessary for transition from box-office bombs to blockbusters. The major factors affecting the critical review score are examined. By iterating the mean-field dynamical equations, we obtain qualitative agreements with simulations and real systems in the dynamical box-office forms, revealing the significant role of competition in understanding box-office dynamics.

Figure 1

A schematic flow diagram of the dynamics. Movie $\alpha$ is introduced at time $t_{\alpha }$. Blue arrows represent the mechanism that draws viewers to watch movies; orange arrows represent viewers posting reviews.

Figure 2

The gross box office of movie $\alpha$ as a function of $Q_{\alpha }$. Parameters: $p=0.05$, $s=1$, and $g=1$. $\langle Q\rangle {}_{\mathrm{o}}$ are obtained with $\rho (Q)~Q^{-2}$ with $Q_{\max }=50$. Inset: The same graph in magnified vertical scale.

Figure 3

$f_{\alpha }(t)$ as a function of $t$ for movies of various quality $Q_{\alpha }$. Parameters: $p=0.05$, $\omega =0.3$, $s=1$, and $g=1$. $\langle Q\rangle {}_{\mathrm{o}}$ are obtained with $\rho (Q)~Q^{-2}$ with $Q_{\max }=50$. Inset: The phase diagram for box-office hits. The critical quality $Q_c^{(\mathrm{I})}$ and $Q_c^{(\mathrm{II})}$ are shown, respectively, by the solid and dashed lines, as a function of $\omega$.

Figure 4

Upper panel: $f_{\alpha }(t)$ and $f_{\beta }(t)$ as a function of $t$ for movie $\alpha$ and $\beta$, respectively, introduced at time $t_{\alpha }$ and $t_{\alpha }+1$ with quality $Q_{\alpha }=Q_{\beta }=1.4\langle Q\rangle {}_{\mathrm{o}}$. Parameters: $p=0.05$, $\omega =0.4$, $s=1$, and $g=1$. $\langle Q\rangle {}_{\mathrm{o}}$ are obtained with $\rho (Q)~Q^{-2}$ with $Q_{\max }=50$. Inset: The same figure with $Q_{\alpha }=1.5\langle Q\rangle {}_{\mathrm{o}}$ and $Q_{\beta }=1.4\langle Q\rangle {}_{\mathrm{o}}$. Lower panel: Detrended $f_{\alpha }(t)$ and $f_{\beta }(t)$, by subtraction of moving average.

Figure 5

The average quality and the number of movies on the recommender list in simulations, as a function of $g$. Parameters: $N=5\times {10}^4$, $p=0.05$, $\omega =0.5$, and $s=1$. Quality of movies is drawn from $\rho (Q)~Q^{-2}$ with $Q_{\max }=50$. Inset: The fraction of relevant reviews as a function of $g$.

Figure 6

The popularity $f(t)$ from simulations (symbols) and mean-field approximation with semiempirical $\langle Q\rangle {}_{\mathrm{o}}$ (dashed lines). Parameters: $N={10}^4$, $p=0.05$, $\omega =0.5$, $s=1$, and $g=1$. Quality of movies is drawn from $\rho (Q)~Q^{-2}$ with $Q_{\max }=50$. Inset: Gross box office as a function of movie quality. The corresponding values of $\langle Q\rangle {}_{\mathrm{o}}$ for analytic, semiempirical, and fitting are 33.7, 27.9, and 15.

Figure 7

The popularity $f(t)$ of 5000 movies on Netflix in the period from October 1998 to December 2005, with $K$, the total number of viewers as indicated in the range. The number of active users is $4.8\times {10}^5$ in this period. The symbol $\times$ indicates the average weekly “opening” of movies.