  We address the issue of the Central Limit Theorem for (both local and global)
empirical measures of diffusions interacting on a possibly diluted
Erd\H{o}s-R\'enyi graph. Special attention is given to the influence of initial
condition (not necessarily i.i.d.) on the nature of the limiting fluctuations.
We prove in particular that the fluctuations remain the same as in the
mean-field framework when the initial condition is chosen independently from
the graph. We give an example of non-universal fluctuations for carefully
chosen initial data that depends on the graph. A crucial tool for the proof is
the use of extensions of Grothendieck inequality.

Probability