# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 4 Discussion
### Stability of the Outputs  
Stability of the outputs is mainly due to our strict stringencies of both defining and identifying the plateaus. It has been popular to rank the significance of outputs by the persistence of plateaus in previous literature. However, as we have argued in the Introduction, if the term \"plateau\" was only loosely defined, which cannot guarantee \"one plateau one topology,\" the related ranks are not surely trustworthy. Although we believe most plateaus in previous literature did have consistent topologies, such an important issue has not been stated, even once.  
In this paper, we suggested a strict stringency that requires not only \"one plateau one topology\", but also a \"best-and-unique\" solution for each relevant resolution. With this stringency, we removed the above suspicions on plateaus, and also rejected unstable results of detection automatically. Here, we raise a simplest example: for an RB25 network, resolution scales exhibited in Figure 11e,f, i.e., \(0.14\leq\alpha\leq 0.25\) for \(\beta=1\), and \(1.27\leq\alpha\leq 1.37\) for \(\beta=2\), only yield unstable divisions of the network. Within these resolution scales, best solutions detected by the Louvain algorithm do not have unique topologies. For instance, Figure 11a-d show four different 4-community divisions for an RB25 network. In the context that nodes of the network are distinguishable, due to the network symmetry, these different divisions all have the same value of \(F_{\alpha}^{\beta}\), and in certain resolution scales, as shown in Figure 11e,f, they can all be detected as best solutions--but not unique. To our viewpoint, these divisions are unstable because different realizations yield different solutions. When the network is more complex, there exist large numbers of such equivalent divisions, none of which are really the best (see our discussion in Section 3.1 on Figure S4). In Figure 11e,f, the central RB5 units are combined with randomly chosen peripheral units, namely, these communities result from random convergences of the heuristic algorithm, and cannot be expected to contain any useful information on the node attributes of the network. Similar phenomena also exist for asymmetrical networks; in both the LFR and the real-world networks, we also observed large numbers of \"best-but-non-unique\" solutions in our junks. Our strategy is designed to automatically reject such random solutions and obtain a clean diagram of plateaus: all our plateaus are stable, and relevant to known community structures. Our selection is inartificial, and the selected results are largely interpretable. In contrast, previous methods did not require the uniqueness of solutions; plateaus for the same networks detected by these methods are much more redundant. Small plateaus emerge within the transitional regions between large plateaus (as the unstable resolution scales shown in Figure 11e,f); see Figure 1 in [5], Figure 2 in [35], Figure 3 in [36], Figures 2 and 3 in [46], and so on. These small plateaus are mostly uninterpretable; one has to rely on the a priori knowledge about the network to select the most interpretable results, and arbitrarily ignore all the rest.  
Figure 11: Unstable community structures for the RB25 network. (**a–d**): Four different 4-community divisions for the RB25 network: the central RB5 unit is randomly combined with one of the peripheral RB5 units. All these divisions have the same value of fitness or modularity, and in certain resolutions scales, they can all be detected as best solutions. However, in the context that nodes of the network are distinguishable, these “best solutions” are non-unique, thus, are not qualified to be classified to any plateau. (**e**,**f**): Unstable resolution scales for the RB25 network detected with \(\beta=1\) and \(\beta=2\). Solid circles exhibit parts of the plateaus shown in Figure 2d,e, and the resolution scales between them only yield _best but non-unique_ solutions, whose numbers of communities are tagged on the corresponding subintervals of resolution.