The case $N=3$ has not been studied, since it never leads to efficiencies higher than 37% [@scholakphd]. Our analysis focuses on structures with $\epsilon>0.9$. Within this reduced set, the number of efficient structures is 3530, 7368, 14280, 5896, 6688 for $N=4$ to $N=8$, respectively (in Fig. \[figsupp:effdistrib\] in Supp. Mat. the probability of generating efficient structures is shown for $N=4-8$). Most of the sets of efficient structures are highly *heterogeneous*, which means that two structures with similar efficiency do not necessarily share any evident common pattern. This structural heterogeneity prevents a straightforward identifications of the geometrical features that are compatible with efficient transport. To uncover these features, we apply the protocol based on complex networks described in Methods.

All the clusters we identify are shown in Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]. A sketch of the most relevant ones is depicted in Fig. \[fig:N4cl12\].

According to the network analysis, systems with $N=4$ and $N=5$ are quite homogeneous, and few geometries are compatible with efficient transport. The configurations obtained are shown in Fig. \[fig:N4cl12\]. In both cases the clusterization algorithm gives two clusters, where the intermediate sites of the less populated cluster are more strongly aligned than the others.

For $N=4$ we label the 