On Observation Time for Recovering Latent Hawkes Networks

TL;DR

This paper establishes the minimal observation time needed to recover latent Hawkes networks, showing it scales logarithmically with the number of entities, and introduces a two-stage estimation method.

Contribution

It provides the first theoretical bounds on observation time for exact network recovery in Hawkes processes, combining upper and lower bounds.

Findings

Logarithmic observation time is sufficient and necessary for network recovery.

A two-stage estimator effectively recovers the network structure.

Concentration bounds are derived from Poisson cluster representations.

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.