Hawkes Processes on Graphons
This work addresses the problem of inferring underlying relations and simulating event sequences for multiple Hawkes processes with heterogeneous event types, which is relevant for researchers working with complex event data.
This paper introduces a framework for modeling multiple multivariate point processes, specifically Hawkes processes, which share an underlying generative mechanism and an uncountable event type space. The model leverages graphons to sample Granger causality graphs of varying sizes, enabling the generation and simulation of event sequences with similar dynamics but different event types.
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.