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We propose a simulation method for multidimensional Hawkes processes based onsuperposition theory of point processes. This formulation allows us to designefficient simulations for Hawkes processes with differing exponentiallydecaying intensities. We demonstrate that inter-arrival times can be decomposedinto simpler auxiliary variables that can be sampled directly, giving exactsimulation with no approximation. We establish that the auxiliary variablesprovides information on the parent process for each event time. The algorithmcorrectness is shown by verifying the simulated intensities with theirtheoretical moments. A modular inference procedure consisting of Gibbs samplersthrough the auxiliary variable augmentation and adaptive rejection sampling ispresented. Finally, we compare our proposed simulation method against existingmethods, and find significant improvement in terms of algorithm speed. Ourinference algorithm is used to discover the strengths of mutually excitationsin real dark networks.

Simulation and Calibration of a Fully Bayesian Marked Multidimensional Hawkes Process with Dissimilar Decays