  We generalise the construction of multivariate Hawkes processes to a possibly
infinite network of counting processes on a directed graph $\mathbb G$. The
process is constructed as the solution to a system of Poisson driven stochastic
differential equations, for which we prove pathwise existence and uniqueness
under some reasonable conditions.
  We next investigate how to approximate a standard $N$-dimensional Hawkes
process by a simple inhomogeneous Poisson process in the mean-field framework
where each pair of individuals interact in the same way, in the limit $N
\rightarrow \infty$. In the so-called linear case for the interaction, we
further investigate the large time behaviour of the process. We study in
particular the stability of the central limit theorem when exchanging the
limits $N, T\rightarrow \infty$ and exhibit different possible behaviours.
  We finally consider the case $\mathbb G = \mathbb Z^d$ with nearest neighbour
interactions. In the linear case, we prove some (large time) laws of large
numbers and exhibit different behaviours, reminiscent of the infinite setting.
Finally we study the propagation of a {\it single impulsion} started at a given
point of $\zz^d$ at time $0$. We compute the probability of extinction of such
an impulsion and, in some particular cases, we can accurately describe how it
propagates to the whole space.
