efficiency loss. This is shown in Fig. \[fig:efflossinactivemod\], where all clusters, apart from $v_8$, behave similarly: the contribution of the inactive modules is particularly important for the most efficient realizations due to the sensitivity of perfect constructive interference. The loss upon pair removal typically ranges from $0.05$ to $0.15$, depending on the initial value of the efficiency. As we suggested from geometrical considerations, $v_8$ is a very noisy cluster and cannot be considered completely composed of a 5-sites backbone plus a triplet. In fact, removal of these three sites causes an efficiency drop up to 60%, which indicates that the triplet in $v_8$ cannot be considered an inactive module.

While the triplets in the first cluster with $N=7$ play a role similar to the pair in $\tau_6$, it is not obvious whether the presence of two pairs in $\tau_{8}^a$ and $\tau_{8}^b$ is necessary or if only one of them is enough to obtain the same effect. In fact, the two pairs show a small degree of collectiveness, which means that one is dominant and the other one has a close-to-negligible effect (Fig. \[fig:efflossinactivemodonebyone\](c-d) in Supp.Mat.). This is confirmed by the fact that the efficiency loss upon removal of the two 