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

Central Limit Theorems for global and local empirical measures of diffusions on Erd\H{o}s-R\'enyi graphs