  We consider a population of $N$ interacting neurons, represented by a
multivariate Hawkes process: the firing rate of each neuron depends on the
history of the connected neurons. Contrary to the mean-field framework where
the interaction occurs on the complete graph, the connectivity between
particles is given by a random possibly diluted and inhomogeneous graph where
the probability of presence of each edge depends on the spatial position of its
vertices. We address the well-posedness of this system and Law of Large Numbers
results as $N\to\infty$. A crucial issue will be to understand how spatial
inhomogeneity influences the large time behavior of the system.
