pairs at the same time is only slightly larger than the sum of efficiency losses upon removal either pair(Fig. \[fig:efflossinactivemodonebyone\](a-b) in Supp. Mat.).

Inactive modules induce eigenvalue shift
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The mechanism behind the influence of the inactive modules on the exciton dynamics can be understood from the distribution of the energy eigenvalues with and without the inactive sites as displayed in Fig. \[fig:lambdashift\]. Because only a single excitation is present in the system at any time, there are $N$ energy eigenvalues to study. Given the weak interaction between the backbone and $k$ pairs or a triplet there are $N-2k$ or $N-3$ eigenstates whose amplitudes are highly localized on the backbone. The amplitudes of the remaining $2k$ or $3$ eigenstates are instead localized on the inactive sites.

The interaction between the backbone and the inactive sites results in a shift of the eigenfrequencies of the former $N-2k$ or $N-3$ eigenstates (denoted by $\mathit{\lambda}_i$ in Fig. \[fig:lambdashift\]), such that their differences are close to integer multiples of a fundamental frequency. With this shift, the excitation is transferred to the output site essentially perfectly after one period of this fundamental frequency. This is true for all the clusters, ranging from the pair and 