On Observation Time for Recovering Latent Hawkes Networks

2026-05-08

AI Summary

This study addresses the recoverability of implicit network structures in stationary Hawkes processes, aiming to determine the minimal observation time required for exact reconstruction. The authors propose a two-stage estimation procedure: first identifying candidate connections from binned and thresholded event data, then refining the network via least squares optimization. Under a sparsity and weak-interaction assumption, they establish—for the first time—a tight information-theoretic lower bound of Θ(log d) on the necessary observation time, where d denotes the number of nodes. This bound is shown to be achievable by combining concentration inequalities derived from the Poisson cluster representation with Jacod’s version of Girsanov’s formula for point processes, thereby characterizing the fundamental limits of network recoverability in Hawkes processes.

Abstract

Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number $d$ of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order $\log d$ is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.

Problem

• latent Hawkes networks
• network recovery
• observation time
• event-based observations
• interacting systems

Innovation

• latent Hawkes networks
• observation time
• network recovery
• Poisson cluster representation
• Fano's inequality