Source-linked AI summary
TiDeH: Time-Dependent Hawkes Process for Predicting Retweet Dynamics
Ryota Kobayashi, Renaud Lambiotte
TL;DR
Predicting the future course of information cascades requires models that use early observations to estimate how activity will evolve, not merely its final size. The paper develops TiDeH, a time-dependent Hawkes process using retweet times, follower information, circadian rhythms, and information aging, and reports systematic improvements over existing approaches across observation windows and prediction scales.
Problem
Information-cascade research needs calibrated models that predict future retweet activity after an observation window, including its temporal evolution rather than only final cascade size.
Method
TiDeH fits a time-dependent Hawkes process to observed retweet times and follower counts, incorporating circadian activity, information aging, and multiple prediction resolutions.
Results
TiDeH performs best across short-to-long observation times and fine-to-coarse prediction scales, with mean and median errors 17.9% and 21.7% smaller than RPP.
Takeaways & Limitations
Modeling retweet diffusion with time dependence and partial network information improves prediction of both future activity profiles and final retweet counts.
Abstract
from arXiv · showhide
Online social networking services allow their users to post content in the form of text, images or videos. The main mechanism driving content diffusion is the possibility for users to re-share the content posted by their social connections, which may then cascade across the system. A fundamental problem when studying information cascades is the possibility to develop sound mathematical models, whose parameters can be calibrated on empirical data, in order to predict the future course of a cascade after a window of observation. In this paper, we focus on Twitter and, in particular, on the temporal patterns of retweet activity for an original tweet. We model the system by Time-Dependent Hawkes process (TiDeH), which properly takes into account the circadian nature of the users and the aging of information. The input of the prediction model are observed retweet times and structural information about the underlying social network. We develop a procedure for parameter optimization and for predicting the future profiles of retweet activity at different time resolutions. We validate our methodology on a large corpus of Twitter data and demonstrate its systematic improvement over existing approaches in all the time regimes.
I. INTRODUCTION
The paper addresses predicting the time evolution of Twitter retweet cascades from early observations, extending cascade-size prediction to future activity profiles. It introduces TiDeH, a time-dependent Hawkes-process model that incorporates network information, circadian variation, and information aging.
- Motivation: Cascade prediction seeks to estimate future information diffusion from observations made during a cascade’s early stage.The problem has applications including content ranking, information presentation, and media campaign management.
- Related work: Existing diffusion-prediction approaches include feature-based classification and calibrated self-exciting point-process models, with the latter reflecting instantaneous events and multiplicative spread.The paper positions TiDeH among methods that combine event-time modeling with partial social-network information.
- Contribution: TiDeH predicts how retweet frequency evolves over time rather than only the cascade’s final size.The model predicts future retweet activity at different temporal resolutions from an observed retweet time series.
- Modeling perspective: TiDeH extends Hawkes processes by allowing the infectiousness parameter to vary daily, capturing circadian rhythms and information aging.Hawkes processes represent self-excitation, where previous events boost future activity through a memory kernel.
- Method: The method fits model parameters using observed retweet times and follower counts, then predicts binned future retweet activity by solving a self-consistent integral equation numerically.The prediction bin width determines the temporal resolution, and final cascade size is recovered by coarse binning or summing future activity.
III. MODELING RETWEET ACTIVITY VIA TIME-DEPENDENT HAWKES PROCESS
TiDeH extends the stationary Hawkes process to model and predict retweet activity using observed retweet times and structural information about the social network. Its time-dependent formulation incorporates infectiousness, follower connectivity, and reaction-time memory while accounting for activity decay over time.
- Data: The study uses 166,076 tweets with retweet times recorded for up to 7 days and limited network-structure information.The dataset also includes follower counts for users associated with original tweets and later retweets.
- Model formulation: TiDeH extends the classical stationary Hawkes process by making the retweet rate depend on previous events.The model describes the probability of a retweet in a small time interval through a time-dependent rate.
- Network structure: Follower counts enter the model through a branching-process representation that gives more importance to highly connected nodes.This weighting is described as analogous to tree-like and heterogeneous mean-field approximations for epidemic spreading on networks.
- Temporal dynamics: The memory kernel represents the probability distribution of a follower’s reaction time after a followee’s tweet.The model combines this reaction-time memory with contributions from previous retweets.
- Model behavior: The infectious rate decreases to zero at sufficiently long times, ensuring that the predicted number of retweets does not diverge.The reported parameters are c0 = 6.49 × 10^-4 (/seconds), s0 = 300 seconds, and θ = 0.242.
- Model interpretation: The instantaneous retweet probability is obtained by multiplying the infectious-rate and memory-effect terms shown in the model illustration.Figure 2 labels these components as the infectious rate, memory effect, and instantaneous probability λ(t).
C. Modeling the infectious rate of a tweet
The infectious rate is estimated from retweet sequences using moving-window maximum likelihood, then modeled with a minimal time-dependent form capturing both circadian oscillations and popularity decay.
- Moving time windows estimate the infectious rate by maximum likelihood under a locally constant-rate assumption.The observation window is set to 4 hours, and the approach is illustrated on retweet sequences.
- Retweet sequences exhibit either slow decay alone or decay combined with circadian oscillations.Figure 3 compares moving-window estimates with fits from the proposed model.
- The proposed time-dependent infectious-rate model represents daily oscillation and a characteristic popularity-decay timescale.Its parameters encode intensity, relative oscillation amplitude, phase, and decay time, with a one-day oscillation period.
- The four model parameters are fitted by minimizing least-square error with constrained oscillation amplitude and decay timescale.The Levenberg–Marquardt algorithm performs the minimization, with −1 < r < 1 and 0.5 < τm < 20 days.
3. Validation of the fitting procedure on synthetic data.
Synthetic-data experiments test whether the fitting procedure reconstructs TiDeH parameters and infectious rates, while examining how observation duration affects estimation accuracy.
- Synthetic-data fitting reconstructs the unobservable infectious rate from simulated retweet sequences.The fitted parameters are compared with ground-truth values over 100 trials.
- 0.0%, 1.9%, 9.6%, and 1.0% are the relative errors for p0, r0, φ0, and τm after 2 days of observation.These results indicate accurate parameter reconstruction for the tested sufficiently long observation period.
- Shorter observation periods reduce estimation accuracy, especially for the phase φ0 and time constant τm below 24 hours.Figure 5 summarizes means and standard deviations across 100 synthetic datasets.
- When r0, φ0, and τm are known, p0 remains accurately estimable from a 1-hour observation window.This restricted case is evaluated in Figure 6.
A. Step 1: Fitting the infectious rate p(t)
The prediction procedure first fits the infectious-rate model from observed retweets, then estimates future activity while accounting for unobserved future self-excitation.
- Three fitting procedures estimate infectious-rate parameters: constant p0, fully fitted TiDeH, and TiDeH with trained shape parameters.The trained variant optimizes r0, φ0, and τm on training data, then estimates p0 for the target sequence using 5-fold cross-validation.
- Future retweet activity is predicted by calculating retweet rate λ(t) from observed retweet times and follower information.Activity is the number of retweets in each prediction bin.
- The conditional expectation of λ(t) incorporates both observed retweets and self-excitation induced during the prediction period.The resulting Volterra integral equation is solved numerically with the trapezoidal method using a 0.1-hour step.
- The model assumes independence between follower counts and retweet times and estimates future follower counts from their observation-window mean.This assumption enters the conditional-expectation formulation through dp.
- Monte Carlo evaluation was not adopted because reasonable retweet-rate estimates require at least 10,000 realizations.The authors identify the computational cost of generating sufficiently large stochastic-process samples as the reason.
C. Effect of the infectious rate models on prediction performance
Prediction performance is evaluated by mean and median absolute error across observation times and prediction resolutions. TiDeH with training is selected as the best-performing method.
- The evaluation uses mean and median absolute error, with predicted and actual retweet activity compared in each time bin through the absolute-error metric.The prediction window ends at Tmax = 168 hours.
- TiDeH outperforms the standard Hawkes process across observation times, including median errors of 8.2 versus 12.6 at T = 1 hour and 1.6 versus 5.6 at T = 1 day.
- Longer observation windows improve prediction accuracy, while training particularly improves performance for short observations below 24 hours.
- The untrained TiDeH model is accurate for sufficiently large observation windows but unsuitable for short observations because parameter fitting deteriorates.
- TiDeH performs significantly better than the standard Hawkes model across prediction resolutions, with error roughly independent of time resolution.
2. Calculate the average number of followers.
Future retweet activity is evaluated by solving the model’s integral equation for the predicted future retweet rate.
- The procedure evaluates the future retweet rate ˆλ(t) by solving the model’s integral equation.
A. Baseline methods for comparison
The study compares TiDeH with several adapted baseline methods, including regression, reinforced Poisson, and branching-process approaches. These baselines were modified because they were originally designed mainly for final cascade-size prediction.
- Previous methods are not directly comparable because they primarily predict final retweet counts, whereas this work predicts retweet activity over time.
- The fourth baseline is used only to evaluate accuracy in predicting the final number of retweets.
- Linear regression: Linear regression predicts the logarithm of popularity from training tweet sequences.
- Linear regression with degree: Linear regression with degree extends regression by incorporating the number of followers into popularity prediction.
- Reinforced Poisson process: The reinforced Poisson process models retweet rate using information aging and a reinforcement mechanism, with parameters fitted by likelihood maximization.The aging function is fγ(t) = t^-γ, and the reinforcement term depends on cumulative retweets.
- SEISMIC: SEISMIC uses a Hawkes-based diffusion model for fitting but a Galton-Watson branching process for predicting future retweets, unlike TiDeH’s Hawkes-based predictor.
B. Prediction results
TiDeH consistently provides the most accurate retweet predictions across observation times, prediction resolutions, and final-retweet counts, outperforming point-process and regression baselines.
- TiDeH performs best across observation times from 1 to 48 hours, followed by RPP, LR-N, and LR.Its accuracy advantage over RPP is systematic, with errors increasing least as observation time decreases.
- 17.9% lower mean error and 21.7% lower median error than RPP are achieved by TiDeH on average.
- TiDeH remains the best predictor from 1-hour to 1-day prediction windows, while error depends only weakly on window size.Error slightly decreases as the prediction window increases.
- Around 30% improvement over SEISMIC and RPP is obtained by TiDeH for predicting the final number of retweets.This comparison uses both mean and median error.
- The evaluated comparisons use 6-hour observation windows for resolution and final-count analyses, and predict activity up to 168 hours in the main time-based setup.
VI. CONCLUSION AND FUTURE WORK
The paper introduces TiDeH, a self-exciting point-process framework for predicting the future time evolution of a tweet’s popularity.
- TiDeH predicts the future time evolution of tweet popularity using a calibrated self-exciting point-process model.