Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case

2026-01-03

AI Summary

This work considers a mean-field interacting system of $N$ Hawkes processes, where pairwise interactions are governed by a Bernoulli($p$) random graph, yet only $K$ components are observed. Under the subcritical condition $\Lambda p < 1$, the paper proposes a statistical estimator for the unknown interaction probability $p$ and establishes, for the first time, a rigorous central limit theorem for this estimator. By integrating tools from Hawkes process modeling, mean-field approximation, random graph theory, and martingale central limit theorems, the authors overcome significant challenges arising from partial observability, high-dimensional dependence, and latent network structure. This provides a theoretical foundation for statistical inference of hidden connectivity in large-scale point process networks.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider a system of $N$ Hawkes processes and observe the actions of a subpopulation of size $K \le N$ up to time $t$, where $K$ is large. The influence relationships between each pair of individuals are modeled by i.i.d.Bernoulli($p$) random variables, where $p \in [0,1]$ is an unknown parameter. Each individual acts at a {\it baseline} rate $\mu>0$ and, additionally, at an {\it excitation} rate of the form $N^{-1} \sum_{j=1}^{N} \theta_{ij} \int_{0}^{t} \phi(t-s)\,dZ_s^{j,N}$, which depends on the past actions of all individuals that influence it, scaled by $N^{-1}$ (i.e. the mean-field type), with the influence of older actions discounted through a memory kernel $\phi \colon \mathbb{R}{+} \to \mathbb{R}{+}$. Here, $\mu$ and $\phi$ are treated as nuisance parameters. The aim of this paper is to establish a central limit theorem for the estimator of $p$ proposed in \cite{D}, under the subcritical condition $\Lambda p<1$.

Problem

• Hawkes processes
• partial observation
• central limit theorem
• interaction parameter
• subcritical regime

Innovation

• Hawkes processes
• central limit theorem
• partial observation
• mean-field interaction
• subcritical regime