hand, if the excitation is completely localized (e.g. at $t=0$ in our case), the IPR adopts its minimal value of 1.

Efficiency Network
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To unravel the structure-dynamics relationship, we apply a set of tools based on complex networks. Originally, these tools had been developed for the characterization of molecular systems [@Rao2004; @Gfeller2007]. However, since these methods are designed to analyze large ensembles of configurations, they prove very useful for our present purposes as they allow a systematic classification of structures which lead to exceptional transport.

We generate a complex network where structures with $\epsilon>0.9$ represent the nodes and a link is placed between them if two structures are geometrically similar independently on the specific dynamics of the excitation. The parameter used to estimate structural similarity depends on the relative distances of the excitable sites of two structures under comparison. The sites are indistinguishable, thus all different permutations of the site labels need to be performed [^1]. In addition, a rotational symmetry around the in-out axis and an additional mirror symmetry has to be taken into consideration. The measure $S$ of similarity between configurations A and B is thus defined as $$S^2=\min\sum^n_{i=1}  \frac{d_i ^2}{n}\ ,$$ where $d_i$ is the difference 