triplet clusters with $N=7$ ($\pi_7$ in Fig. \[fig:lambdashift\]) to the clusters with two pairs in the $N=8$ case ($\tau_{8}^a$ and $\tau_{8}^b$ in Fig. \[fig:lambdashift\]), what strongly suggests that this mechanism [@Mostarda2013] works independently of the system size.

![Shift of the first backbone eigenvalue upon removal of inactive modules. In systems with both $N=7$ (top row) and $N=8$ (bottom row) the removal of the inactive sites causes a shift in only the first eigenvalue. In the former case, the mechanism is similar for both the pair and the triplet.[]{data-label="fig:lambdashift"}](eigshift_7.pdf){width="44.00000%"}

Robustness
----------

The analysis presented so far shows the emergence of two classes of geometrical and dynamical behavior, characterized by an arrangement into active and inactive modules. In the following, we explore this separation with respect to the robustness properties of the various clusters.

![image](robustn_8.pdf){width="99.00000%"}

Transport robustness is probed by random displacements of the individual sites of a structure. With displacements confined to a cube of side $0.05 \  r_0$ centered around the original position of the site, $\mathit{\Delta} \epsilon_{\text{rand}}$ is calculated for each structure as the difference between the original efficiency and the average efficiency obtained from 1000 site-randomizations. In this scheme, structures are kept rigid which corresponds to the assumption 