Recently, a conceptually new approach for analyzing gene networks, the Functional Influence Network (FIN) was presented. The FIN approach uses the measured performance of a given cellular function under different multi-perturbations, to identify the main functional pathways and interactions underlying its processing. Here we present and study an iterative, extended version of FIN, the Functional Influence Network Extractor (FINE), which is specifically geared towards the accurate analysis of sparse cellular systems. We employ it to study a conceptually fundamental question of practical importance--how well should we know the system studied (such that we can predict its performance) so that we can understand its workings (i.e., chart its underlying functional network)?