Hawkes Processes on Graphons

TL;DR

This paper introduces a nonparametric graphon-based framework for modeling multiple multivariate Hawkes processes with shared underlying structures, enabling inference of relations and simulation of event sequences.

Contribution

It proposes a novel graphon-based Hawkes process model that captures shared dynamics across processes and introduces a hierarchical optimal transport learning method.

Findings

The model effectively infers underlying relations among Hawkes processes.

It can generate realistic event sequences with similar dynamics.

The approach is validated both theoretically and experimentally.

Abstract

We propose a novel framework for modeling multiple multivariate point processes, each with heterogeneous event types that share an underlying space and obey the same generative mechanism. Focusing on Hawkes processes and their variants that are associated with Granger causality graphs, our model leverages an uncountable event type space and samples the graphs with different sizes from a nonparametric model called {\it graphon}. Given those graphs, we can generate the corresponding Hawkes processes and simulate event sequences. Learning this graphon-based Hawkes process model helps to 1) infer the underlying relations shared by different Hawkes processes; and 2) simulate event sequences with different event types but similar dynamics. We learn the proposed model by minimizing the hierarchical optimal transport distance between the generated event sequences and the observed ones, leading to a novel reward-augmented maximum likelihood estimation method. We analyze the properties of our model in-depth and demonstrate its rationality and effectiveness in both theory and experiments.