Source-linked AI summary
A Bayesian approach for predicting the popularity of tweets
Tauhid Zaman, Emily B. Fox, Eric T. Bradlow
TL;DR
The paper asks whether tweet popularity, measured by retweet evolution, can be predicted very early from limited retweet and network information. It develops a Bayesian model of retweet evolution and reports accurate forecasts of eventual retweet totals from only a small early fraction of the path, with implications for studying broader content spread.
Problem
The paper addresses the problem of predicting a tweet’s popularity from its retweet-time path, including the eventual total, rather than only pairwise retweets or coarse count intervals.
Method
The model uses Bayesian inference with early retweet times, retweet-graph depth, follower-network summaries, and related retweet information.
Results
The median APE (MAPE) is below 40% for observation fractions from 10% to 100%, while the median time for total-retweet predictions to enter the 90% posterior credible interval is 3 minutes.
Takeaways & Limitations
Early predictions can identify popular tweets before they receive many retweets and may support applications involving impressions and the spread of broader ideas, memes, or trends.
Takeaways & Limitations
Tweets with initially rapid retweet growth followed by substantial slowing deviate from the log-normal reaction-time model and produce larger prediction errors.
Abstract
from arXiv · showhide
We predict the popularity of short messages called tweets created in the micro-blogging site known as Twitter. We measure the popularity of a tweet by the time-series path of its retweets, which is when people forward the tweet to others. We develop a probabilistic model for the evolution of the retweets using a Bayesian approach, and form predictions using only observations on the retweet times and the local network or "graph" structure of the retweeters. We obtain good step ahead forecasts and predictions of the final total number of retweets even when only a small fraction (i.e., less than one tenth) of the retweet path is observed. This translates to good predictions within a few minutes of a tweet being posted, and has potential implications for understanding the spread of broader ideas, memes, or trends in social networks.
1. Introduction.
This paper predicts tweet popularity from the evolving time path of retweets, using early timing and retweeter-network information. It targets forecasts of eventual retweet counts within minutes, distinguishing its task from pairwise retweet prediction and coarse popularity intervals.
- The paper models tweet popularity through the time path of retweets, where retweeting forwards a tweet through Twitter’s follower graph.Tweets can contain up to 140 characters, images, or links.
- A Bayesian model uses early retweet times, other tweets’ retweets, and summaries of follower graphs to predict a tweet’s eventual total retweets.
- The paper is motivated by early forecasting, aiming to make predictions sometimes within minutes of a tweet being posted.
- The prediction goal is to estimate the entire retweet path and final retweet count, rather than only whether a user pair will retweet or which coarse count interval applies.
- Exploratory data analysis characterizes retweet counts and timescales and guides the formal probabilistic model.
2. Data overview.
The study analyzes 52 Twitter retweet graphs across varied topics, characterizing their size, lifetime, depth, reaction times, and structural predictors. These exploratory patterns motivate a model that uses retweet timing and graph information to predict popularity.
- Data description: The dataset contains 52 root tweets spanning music, politics, and everyday events, with at most 1,800 retweets per tweet.Tweets were at least one week old when collected through the Twitter Search API.
- Graph representation: Retweet graphs contain the root user and retweeters, with directed edges identifying who retweeted whom and total size measured by the number of retweets.The graph observed by time t is G_x(t), and its final retweet count is M_x.
- Size and lifetime: The 52 tweets received 21–1260 retweets, while median retweet times ranged from four minutes to three hours, usually under one hour.Final retweets could occur over several hours or days, making the median a more stable lifetime measure.
- Size and lifetime: Median retweet time was not associated with eventual retweet count: Pearson correlation was −0.12 (p-value = 0.49) and Kendall tau was 0.03 (p-value = 0.84).The result suggests that simple central summaries are insufficient for predicting final popularity.
- Graph depth: Most retweeters were one hop from the root: 11,882 were one hop away versus 314 more than one hop away across all graphs.This depth structure supports early prediction because later-hop retweets are relatively uncommon.
- Reaction times: Reaction time is the elapsed time between a user's parent retweet and that user's retweet, and is modeled as an i.i.d. log-normal variable within each root tweet.The log-normal parameters vary by root tweet to capture content-dependent heterogeneity.
- Model assumptions: The conditional-independence assumption is supported in this dataset by cycle-forming follower edges comprising less than 1% of all possible follower edges.The authors note that other networks may require generalizations.
- Graph structure: The follower-count and depth regression estimates were β̂_0 = 1.99, β̂_f = −0.79, and β̂_d = −4.31, with all corresponding p-values below 0.001.The negative depth coefficient indicates lower retweet likelihood farther from the root user.
3. Retweet model.
The paper models retweet graphs as recursive processes in which users generate reaction times and numbers of children, then uses a hierarchical Bayesian model to predict partially observed retweet paths and final graph structure.
- Generative process: The generative process starts with a root tweet and recursively assigns each retweeting user a reaction time and an independent number of children.The process ends when graph leaves have no children.
- Generative process: The number of children depends on node depth, and the model typically learns fewer children for nodes at greater depths.
- Reaction-time model: Reaction times are modeled with tweet-specific log-normal distributions whose parameters receive hierarchical Bayesian priors and share information across tweets.The graphical model represents this cross-tweet shrinkage.
- Graph-structure model: Retweet probabilities are modeled independently across potential followers while allowing follower count and retweet depth to influence eventual retweeters through coefficients shared across root tweets.
- Prediction: For prediction tweets, the model treats final degrees as missing, incorporates only retweets observed before the prediction time, and marginalizes over unobserved graph structure.
- Prediction: Temporal retweet observations are dependent through the latent final graph, allowing them to inform predictions of final graph structure and total retweet count.The likelihood has no closed form, so missing final degrees are imputed using MCMC-based posterior computation.
4. Results.
The Bayesian retweet model produces accurate early forecasts of total retweet counts, using partial retweet histories and retweet-graph information. Across prediction tweets, its errors remain below 40% from 10% to 100% observation, and it outperforms the benchmark models.
- Posterior analysis: The model’s posterior means include α = 7.42, compared with a mean maximum-likelihood estimate of αx = 7.31, and the β parameters’ 90% credible intervals exclude 0.The results indicate that the β parameters contribute to the model’s predictive power.
- Retweet prediction results: At a 10% observation fraction, the median time for total-retweet predictions to enter the 90% posterior credible interval was 3 minutes.Predictions made within a few minutes were relatively close to the true eventual retweet count.
- Retweet prediction results: The model’s median absolute percentage error (MAPE) was below 40% for observation fractions from 10% to 100%.At 10% observation, the average retweet time was 4.4 minutes; across ten random partitions, 10% MAPE ranged from 20% to 36%, averaging 28%.
- Comparison with benchmark models: The model outperformed the follower-count regression model, the Szabo and Huberman model, and the dynamic Poisson model on final and remaining retweet-count MAPE.The follower-count regression model had a MAPE of 65%, while the other comparisons are summarized in Figure 12.
- Comparison with benchmark models: At 10% observation, the model’s MAPE was 29%, compared with 80% for the strawman model.The strawman error decreased initially as more retweets were observed, then increased again.
- Comparison with benchmark models: Follower information and retweet-graph depth are important elements for accurately predicting retweets.The comparison of model fit uses average log-likelihood and deviance information criterion, with the proposed model fitting significantly better than the strawman.
5. Model extension opportunities.
The model has several extension opportunities and identified specific conditions that limit prediction accuracy. These include unobserved future retweeters, reaction-time deviations from log-normality, and omitted tweet side information.
- Unobserved future retweeters: Current forecasts exclude retweets from future retweeters not yet observed, although incorporating them could improve predictions with more follower-graph data.The existing approach is effective when retweet graphs are shallow and retweets occur quickly; decreasing retweet probability with depth may reduce sensitivity to future-retweeter inaccuracies.
- Reaction-time modeling: Four tweets had very large errors at 40% observation, with rapidly increasing retweets followed by substantial slowing.Their retweet counts ranged from 73 to 608, and this behavior deviated from the log-normal reaction-time model.
- Reaction-time modeling: The high-error tweets combined short median reaction times with large ∆x values, indicating poor fit to the log-normal distribution.Figure 14 compares median reaction time with ∆x and marks the high-error tweets with triangles.
- Side information: Predictions primarily use retweet timing, retweet-graph depth, and follower counts, while time of day and tweet content could be added as covariates.The paper identifies posting time, topic, and trending status as potentially useful side information.
- Current performance: MAPE was below 40% when at least 10% of total retweets were observed, corresponding for most tweets to an average error below 40% within five minutes.These results define the current model’s demonstrated early-prediction performance.
6. Conclusion.
The model can use retweeter-network size and graph depth to estimate potential tweet viewers, supporting applications involving early popularity assessment. Its single-tweet formulation can also serve as a foundation for modeling the spread of broader ideas across many tweets.
- Applications: Retweeter-network size and depth from the source tweet allow prediction of a tweet’s potential viewers.The paper connects this accuracy to possible applications such as treating tweets as potential sources of display-ad impressions and detecting popularity while tweets are actively retweeted.
- Broader idea spread: A single-tweet model can be parallelized across large tweet collections and extended toward understanding the speed and magnitude of meme and trend popularity.The proposed broader-ideas direction involves multiple tweets from multiple users.
APPENDIX A: DETAILS OF MCMC SAMPLER
The sampler obtains posterior distributions for model parameters using a Metropolis-within-Gibbs procedure conditioned on observed reaction times and the remaining parameters.
- Posterior sampling: The MCMC procedure samples each model parameter from its conditional distribution given observed reaction times and all other parameters.The parameter set is Θ, and the sampler updates each parameter γ conditional on S, M_T, and Θ−γ.
A.1. Retweet graph structure parameters.
The sampler uses parameter-specific conditional updates for the retweet-graph structure parameters, combining direct Gibbs sampling with Metropolis–Hastings proposals where needed.
- Coefficient updates: The conditional distribution for the regression coefficients is multivariate normal, allowing direct Gibbs sampling.The coefficients β0, βF, and βd have normal priors with mean 0 and standard deviation σβ = 100.
- Variance update: The conditional distribution for σ2_b is inverse-gamma, so it can be sampled directly.Its prior inverse-gamma shape and scale parameters are both 0.5.
- Metropolis–Hastings updates: Other graph-structure parameters are updated with Metropolis–Hastings proposals, including proposals for M^x_j and logit(b_x).The supplied sampler description also specifies normal and binomial proposal distributions for selected parameters.
A.2. Retweet time parameters.
The retweet-time model uses Bayesian priors and conditional distributions for global and tweet-specific parameters. Some parameters are sampled directly, while others use Metropolis–Hastings updates.
- The model assigns α a diffuse normal prior with mean 0 and standard deviation 100.
- The prior for log(a_τ) is normal with mean 0 and standard deviation 10.
- The prior for b_τ is gamma with shape parameter k_b = 1 and scale parameter θ_b = 500.
- Conditional distributions for α_x can be normal for training tweets, while sampling a_τ uses random-walk Metropolis–Hastings with proposal standard deviation 0.2.
Parameters αx, τ x.
For prediction tweets, the model updates tweet-specific parameters using observed retweets and Bayesian conditional distributions. Direct inverse-gamma sampling is available in some cases, while prediction settings require Metropolis–Hastings.
- For a prediction tweet with n observed retweets, the conditional distribution of α_x is specified separately from the training-tweet distribution.
- The prior for (τ_x)^2 is inverse-gamma with shape and scale parameters a_τ and b_τ.
- For prediction tweets, (τ_x)^2 is updated with Metropolis–Hastings using an inverse-gamma proposal distribution.
APPENDIX B: DISTRIBUTED IMPLEMENTATION OF MCMC SAMPLER
The MCMC sampler separates global parameters from tweet- and user-specific variables. Its local-variable updates can be parallelized across tweets and tweet/user pairs, enabling a data-parallel implementation.
- The sampler’s variables are global or local, with the main computational burden arising from potentially thousands or millions of local variables.
- The sampling steps for tweet/user-pair variables can be performed in parallel across all pairs (x,j).
- Sampling α_x and τ_x must be sequential within each tweet but can run in parallel across tweets.
- The sampler updates global parameters first, then tweet-specific parameters in parallel across tweets, followed by tweet/user-pair parameters in parallel.
SUPPLEMENTARY MATERIAL
The supplementary material provides retweet time-series data for the root tweets and the training/prediction partitions used in the analysis.
- The supplement contains retweet time-series data and the different training and prediction partitions used for the paper’s analysis.