We observe the actions of a $K$ sub-sample of $N$ individuals up to time $t$ for some large $K\le N$. We model the relationships of individuals by i.i.d. Bernoulli($p$)-random variables, where $p\in (0,1]$ is an unknown parameter. The rate of action of each individual depends on some unknown parameter $μ> 0$ and on the sum of some function $φ$ of the ages of the actions of the individuals which influence him. The parameters $μ$ and $φ$ are considered as nuisance parameters. The aim of this paper is to obtain a central limit theorem for the estimator of $p$ that we introduced in \cite{D}, both in the subcritical and supercritical cases.