  We consider a population of Hawkes processes modeling the activity of $N$
interacting neurons. The neurons are regularly positioned on the segment
$[0,1]$, and the connectivity between neurons 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 through a spatial kernel. The
main result of the paper concerns the longtime stability of the synaptic
current of the population, as $N\to\infty$, in the subcritical regime in case
the synaptic memory kernel is exponential, up to time horizons that are
polynomial in $N$.
