  We propose a new Kalikow decomposition and the corresponding Perfect
Simulation algorithm for continuous time multivariate counting processes, on
potentially infinite networks. We prove the existence of such a decomposition
in various cases. This decomposition is not unique and we discuss the choice of
the decomposition in terms of algorithmic efficiency. We apply these methods on
several examples: linear Hawkes process, age dependent Hawkes process,
exponential Hawkes process.
