Characterizing and Modeling the Dynamics of Online Popularity

Online popularity has an enormous impact on opinions, culture, policy, and profits. We provide a quantitative, large scale, temporal analysis of the dynamics of online content popularity in two massive model systems: the Wikipedia and an entire country’s Web space. We find that the dynamics of popularity are characterized by bursts, displaying characteristic features of critical systems such as fat-tailed distributions of magnitude and interevent time. We propose a minimal model combining the classic preferential popularity increase mechanism with the occurrence of random popularity shifts due to exogenous factors. The model recovers the critical features observed in the empirical analysis of the systems analyzed here, highlighting the key factors needed in the description of popularity dynamics.

Figure 1

Time series of indegree $k$ and its logarithmic derivative $\Delta k/k$ for the Wikipedia topic page about the artist Jennifer Hudson. Topics typically experience a burst in their early life. Here we observe later fluctuations as well. Jennifer Hudson became popular through a television show leading to her first burst. Another occurred when she won an Academy Award; degree popularity doubled as many other pages linked to the article (inset). The size of each circle shows another popularity measure; it is proportional to the log-derivative of the number of times the article is revised. The article receives more edits when it attracts more links.

Figure 2

(a)–(c) Distributions of popularity burst size. The gray areas highlight the events for which $\Delta k>k$ (hence $\Delta k/k>1$). Maximum likelihood methods [34] in conjunction with the Kolmogorov-Smirnoff (KS) statistic rule out log-normal fits. In each case the KS statistic suggests that the power-law curve is the better fit for the tail. For the distribution of $\Delta k/k$ in Wikipedia (a) the parameters are $\alpha =2.6$ for the exponent of the power law, with a lower cutoff of 12 and a KS statistic of 0.005. For the Web (b) we find $\alpha =1.9$ for the exponent of the power law, with a lower cutoff of 42 and a KS statistic of 0.007. For the distribution of $\Delta s/s$ (c) the parameters are $\alpha =2.1$ with lower cutoff 90 and KS statistic 0.007. The slopes of the best fit power laws are shown as a guide to the eye. These behaviors are consistent across a wide range of temporal resolutions, as observed by using time units from a day to a year. (d) Distribution of the time interval $\Delta t$ between consecutive indegree bursts of Wikipedia articles. We consider bursts such that $\Delta k/k>1$ after January 1st, 2003. The three curves correspond to different time resolutions of months, weeks, and days, aligned on the $x$ axis for ease of visualization. As we increase the resolution, the tail of the distribution extends further, an indication that the cutoff is a finite size effect. As a guide to the eye we show a power law $P(\Delta t)\sim (\Delta t{)}^{-\beta }$ with $\beta \approx 0.8$.

Figure 3

(a) Comparison of the empirical burst size distributions with what would be expected from a preferential attachment (PA) process. Extensive numerical tests and maximum likelihood fitting [34] show that PA generates an approximately log-normal distribution (defined inside the gray area) inconsistent with the long tail observed in the empirical data. (b) The empirical interburst time distributions overlap when time is expressed in terms of the same unit (in the figure, the common time unit is one day). The distribution generated by PA is much narrower and fits an exponential $P(\Delta t)\sim e^{-\Delta t/\tau }$ with $\tau =0.8$. (c),(d) The rank-shift model, despite its simplicity, reproduces quite well the distributions of both event size (c) and interevent time (d).

Figure 4

Rank-shift model. (a),(b) Indegree distribution: $\delta =1$ (a) and $\delta =1.5$ (b). (c) Comparison of the distribution of popularity bursts for the ranking model [30] (circles) and a stylized model built upon the simple assumptions of growth described in the text. (d) Comparison of the distribution of popularity bursts with the expected slope derived by assuming that nodes are reranked at most once.