  We consider a population of $N$ individuals, of which we observe the number
of actions as time evolves. For each couple of individuals $(i,j)$, $j$ may or
not influence $i$, which we model by i.i.d. Bernoulli$(p)$-random variables,
for some unknown parameter $p\in (0,1]$. Each individual acts autonomously at
some unknown rate $\mu>0$ and acts by mimetism at some rate depending on the
number of recent actions of the individuals which influence him, the age of
these actions being taken into account through an unknown function $\varphi$
(roughly, decreasing and with fast decay). The goal of this paper is to
estimate $p$, which is the main charateristic of the graph of interactions, in
the asymptotic $N\to\infty$, $t\to\infty$. The main issue is that the mean
field limit (as $N \to \infty$) of this model is unidentifiable, in that it
only depends on the parameters $\mu$ and $p\varphi$. Fortunately, this mean
field limit is not valid for large times. We distinguish the subcritical case,
where, roughly, the mean number $m_t$ of actions per individual increases
linearly and the supercritical case, where $m_t$ increases exponentially.
Although the nuisance parameter $\varphi$ is non-parametric, we are able, in
both cases, to estimate $p$ without estimating $\varphi$ in a nonparametric
way, with a precision of order $N^{-1/2}+N^{1/2}m_t^{-1}$, up to some
arbitrarily small loss. We explain, using a Gaussian toy model, the reason why
this rate of convergence might be (almost) optimal.
