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Common Drivers in Sparsely Interacting Hawkes Processes

Alexander Kreiss, Enno Mammen, Wolfgang Polonik

TL;DR

This work tackles high-dimensional, time-continuous relational networks modeled by multivariate Hawkes processes with global covariates and actor-specific covariates, where the network structure is unknown and sparsity is assumed to prevent explosion. It develops a partially penalized least-squares estimation framework, augmented by a de-biasing procedure to enable valid inference for the global covariate effects, and it provides vertex-level oracle-type guarantees for the network parameters. The authors establish convergence rates for the first-stage estimators, prove asymptotic normality for the de-biased global parameters, and derive oracle inequalities for the final stage estimators under random compatibility conditions. Empirical results via simulations and an R package demonstrate that incorporating common drivers improves network recovery and that de-biasing yields more reliable inference, albeit with sparsity requirements for the fast-rate results. Overall, the paper advances scalable estimation and inference for nonstationary, high-dimensional Hawkes networks with covariates, offering practical tools and theoretical guarantees for applications in relational event data.

Abstract

We study a multivariate Hawkes process as a model for time-continuous relational event networks. The model does not assume the network to be known, it includes covariates, and it allows for both common drivers, parameters common to all the actors in the network, and also local parameters specific for each actor. We derive rates of convergence for all of the model parameters when both the number of actors and the time horizon tends to infinity. To prevent an exploding network, sparseness is assumed. We also discuss numerical aspects.

Common Drivers in Sparsely Interacting Hawkes Processes

TL;DR

This work tackles high-dimensional, time-continuous relational networks modeled by multivariate Hawkes processes with global covariates and actor-specific covariates, where the network structure is unknown and sparsity is assumed to prevent explosion. It develops a partially penalized least-squares estimation framework, augmented by a de-biasing procedure to enable valid inference for the global covariate effects, and it provides vertex-level oracle-type guarantees for the network parameters. The authors establish convergence rates for the first-stage estimators, prove asymptotic normality for the de-biased global parameters, and derive oracle inequalities for the final stage estimators under random compatibility conditions. Empirical results via simulations and an R package demonstrate that incorporating common drivers improves network recovery and that de-biasing yields more reliable inference, albeit with sparsity requirements for the fast-rate results. Overall, the paper advances scalable estimation and inference for nonstationary, high-dimensional Hawkes networks with covariates, offering practical tools and theoretical guarantees for applications in relational event data.

Abstract

We study a multivariate Hawkes process as a model for time-continuous relational event networks. The model does not assume the network to be known, it includes covariates, and it allows for both common drivers, parameters common to all the actors in the network, and also local parameters specific for each actor. We derive rates of convergence for all of the model parameters when both the number of actors and the time horizon tends to infinity. To prevent an exploding network, sparseness is assumed. We also discuss numerical aspects.

Paper Structure

This paper contains 30 sections, 18 theorems, 282 equations, 10 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Estimation
  3. De-biasing in a Hawkes Model
  4. Results
  5. First Stage Consistency
  6. Second Stage: De-biasing
  7. Third Stage Estimation
  8. Empirical Results
  9. Implementation
  10. Tuning Parameter Choice
  11. Simulation Study
  12. Data Generating Process
  13. Simulation Results
  14. Conclusion
  15. Algorithmic and computational considerations
  16. ...and 15 more sections

Key Result

Theorem 3.2

Let $a_n,b_n,d_{n,i},e_n\in\mathbb{R}$ for $n\in\mathbb{N}$ and $i=1,...,n$, and let $\widetilde{b}_n\geq\max(b_n,e_n)$. Suppose that $d_{n,i}\leq\omega_i$ and that there is a number $L\in(0,\infty)$ such that Denote If we restrict to the parameter space $\widetilde{\mathcal{H}}_n$, we have under (PE1) on the event that

Figures (10)

  • Figure 1: Example of causal DAG for a multivariate process $N=(N_1,...,N_4)$ and confounder $X$.
  • Figure 2: Dots show the mean value of the global covariate and the solid shows a the actual values used in $X_{n,i}^{(1)}$.
  • Figure 3: Histograms of estimators for $\beta_0$, the vertical line shows the true value and the normal distribution is chosen to align the data for illustration.
  • Figure 4: The top left panel shows the true interactions. The other panels show (in the indicated scenarios) those $15$ edges that receive on average the highest weights over all $N$ simulations. Edge thickness is proportional to their weight and vertex size is proportional to the value of $\alpha_{n,i}$ (true or estimated).
  • Figure 5: Percentage of detections per edge in all considered scenarios.
  • ...and 5 more figures

Theorems & Definitions (45)

  • Remark 2.1
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5
  • Remark 3.6
  • Corollary 3.7
  • Theorem 3.8
  • Lemma 3.9
  • ...and 35 more