The Ancestor Hawkes Process with an Application to Group Chat Data

Technical Summary: The Ancestor Hawkes Process with an Application to Group Chat Data

Problem Statement
The standard multivariate Hawkes process is a widely used model for point process data characterized by clustering and bursts, with applications ranging from seismology to finance. In the classic formulation, the conditional intensity of events in a specific dimension is driven by a background rate and the cumulative influence of all past events. A critical limitation of this standard model is its assumption that every event exerts an equal influence on the process intensity based solely on the dimension in which it occurred, irrespective of the event's "branching" history.

In many real-world scenarios, such as group chat messaging, the context of an event matters significantly. Specifically, an event that initiates a new conversation (an "immigrant" event) may have a different impact on future interactions than an event that is a reply within an existing thread (a "triggered" event). The classic Hawkes process cannot distinguish between these two types of events, treating all messages from a specific sender as having identical influence magnitudes. This leads to a structural inability to capture nuanced dynamics, such as the difference in response propensity to a new topic versus a reply to an ongoing discussion.

Methodology
To address this limitation, the authors propose the Ancestor Hawkes process, a novel extension of the multivariate Hawkes model that incorporates the latent branching structure of events directly into the parameter architecture.

• Model Definition: The model introduces two distinct influence magnitude matrices, $K$ and $L$.

  • Matrix $K$ governs the influence of immigrant events (exogenously generated events, $B_i=0$) on other dimensions.
  • Matrix $L$ governs the influence of triggered events (events generated by previous events, $B_i > 0$) on other dimensions.
  • The conditional intensity for dimension $m$ at time $t$ is defined as:
    $$\lambda_m(t) = \mu_m(t) + \sum_{i: t_i < t, B_i=0} K_{d_i \to m} g_{d_i \to m}(t - t_i) + \sum_{i: t_i < t, B_i > 0} L_{d_i \to m} h_{d_i \to m}(t - t_i)$$
  • Here, $g$ and $h$ represent the influence kernels for immigrant and triggered events, respectively. The model assumes that immigrant and triggered events may follow different kernel shapes and magnitudes.

• Stability: The authors establish that the process is stable (subcritical) if the spectral radius of the triggered-to-triggered influence matrix $L$ is less than 1 ($\rho(L) < 1$). This differs from the standard Hawkes process, where stability depends on the spectral radius of the single influence matrix.

• Estimation: Since the branching variables ($B_i$, indicating whether an event is an immigrant or triggered) are unobserved in real data, the authors develop a Bayesian Gibbs sampler for posterior inference.

  • The sampler jointly updates the model parameters ($\mu, K, L, \theta_g, \theta_h$) and the latent branching variables $B$.
  • A key technical challenge addressed is that, unlike in the classic Hawkes sampler where branching variables can be updated independently, the Ancestor Hawkes variables are mutually dependent because assigning an event as immigrant or triggered changes the likelihood of future events. The authors resolve this by updating branching variables in a reverse-time sweep (from $N$ down to 1), conditioning on the current state of all other variables.

Key Contributions and Results

1. Theoretical Extension: The paper formalizes the Ancestor Hawkes process, distinguishing between exogenous starts and reply-driven influence. It provides stability conditions and a rigorous Bayesian inference framework.
2. Simulation Studies:
   • Constant Background Rates: In simulations with known ground truth, the classic Hawkes model was shown to produce biased estimates, effectively averaging the distinct influence magnitudes of immigrant and triggered events (blurring $K$ and $L$). In contrast, the Ancestor Hawkes model successfully recovered the true separate matrices $K$ and $L$, even when the influence kernels were identical.
   • Time-Varying Background Rates: A second simulation mimicking the real-world application (9 dimensions, seasonal background rates) confirmed that the inference procedure can accurately recover the generating influence matrices with high correlation (0.989 for $K$ and 0.995 for $L$).
3. Real-World Application (Group Chat Data):
   • The model was applied to a dataset of 10,705 messages from a 9-person group chat over 3.5 years. Only sender identity and timestamps were used; message content was not analyzed, ensuring privacy.
   • Findings: The Ancestor Hawkes model revealed distinct response preferences. For instance, an immigrant message (initiating a topic) from a specific user elicited significantly more responses from the group than a triggered message (a reply) from the same user.
   • Comparison: The classic Hawkes model failed to capture this distinction, with its posterior estimates lying between the $K$ and $L$ distributions of the Ancestor model.
   • Predictive Performance: Posterior predictive checks using summary statistics (mean upper-tail inter-event time, lag-1 autocorrelation, and Ripley's K-statistic) demonstrated that the Ancestor Hawkes model provided a significantly better fit to the observed clustering structure than the classic model.

Significance and Claims
The authors claim that the Ancestor Hawkes process offers a necessary refinement for modeling multivariate event streams where the "origin" of an event (exogenous vs. endogenous) fundamentally alters its impact on future activity.

• Privacy-Conscious Analysis: A significant practical contribution is the ability to infer latent interaction structures and individual response preferences using only metadata (sender and time), without accessing sensitive message content. This makes the approach suitable for privacy-conscious applications in social media analytics.
• Capturing Latent Dynamics: The paper argues that the distinction between "spontaneous starts" and "replies" is a genuine feature of conversational dynamics that standard models miss. By separating these mechanisms, the model provides a more accurate representation of group interaction structures, such as identifying who initiates conversations versus who primarily responds.
• General Applicability: While demonstrated on group chat data, the authors note the model's relevance to other multivariate streams, such as contact-center queues, organizational emails without reply metadata, and re-share cascades, where separating exogenous load from reply-driven coupling is valuable.

The paper concludes modestly, acknowledging that while the model captures these nuances effectively, future work could incorporate additional covariates (like explicit reply metadata or message types) and develop faster estimation procedures for larger-scale multivariate settings.