# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### On the Hierarchical Ravas-Barabasi (RB) Networks
To fix this problem, we take \(\beta\geq 2\). It turns out, that with \(\beta\geq 2\) our method discovers all community levels including the intermediate ones for the RB networks; see Figures 2e and 3f, Supplementary Figures S1c and S2b for the plateaus obtained with \(\beta=2\). With \(\beta>2\), we simply obtain similar results. As observed, for RB networks with no more than 5 levels, our method accurately and exclusively recovers the divisions suggested in Table 1. As for the potential variants due to relaxed stringencies, in some realizations we did detect some of them as \"local best solutions.\" Yet globally (i.e., over all realizations), these variants are not best-and-unique, thus, they are discarded by our filtering strategy. In contrast, the divisions suggested in Table 1 perform more robustly, and our filtering strategy accurately hits on these \"robust divisions\". On the other hand, our results also confirm that the scaling factor \(\beta\) does rescale the community fitness function effectively, which facilitates our detections on multilevel community structures. Yet our method also has a limit. We notice that with the increase of \(\beta\), the resolution ranges for different community levels span differently: the highest and lowest levels always occupy a majority of the resolution scales, while intermediate levels only emerge within a limited region (when \(\beta=2\), it displays as \(1<\alpha<1.6\)). For RB networks deeper than 5 community levels, some intermediate levels will be compressed and cannot be detected by our method. For example, for an RB15625 network, plateaus for the second and fifth community levels, which are expected to represent 1250 and 22 communities, are both missing, see Supplementary Figure S2d. Within the resolution scales where these community levels are expected to emerge, the Louvain algorithm fails to converge to a best-and-unique solution. Further increasing \(\beta\) does not solve this problem. As shown in Supplementary Figure S3, with the increase of \(\beta\), resolution scales for intermediate community levels do not expand remarkably. Within the scope of this paper, the limit of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**RB5\({}^{n}\) Networks**} & \multicolumn{5}{c}{**Community Levels (\(m\))**} \\ \cline{2-6}  & **1 (Lowest)** & **2** & **3** & **4** & **5** & **6** \\ \hline RB25 (\(n=2\)) & 6 & 1 & / & / & / & / \\ RB125 (\(n=3\)) & 30 & 10 & 1 & / & / & / \\ RB625 (\(n=4\)) & 150 & 50 & 14 & 1 & / & / \\ RB3125 (\(n=5\)) & 750 & 250 & 70 & 18 & 1 & / \\ RB15625 (\(n=6\)) & 3750 & 1250 & 350 & 90 & 22 & 1 \\  & & (undetectable) & & & (undetectable) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numbers of communities on different community levels in our proposed divisions for the RB networks. Here, \(m=1\) stands for the lowest community level, which divides the network into smallest communities that cannot split further; \(m\) is restricted to be no larger than \(n\). Numbers in this table all follow our proposed formula: (\(4m+2\)) \(\times\)\(5^{n-m-1}\). Our method detects all levels of communities when the network is not extremely large (\(n\leq 5\)). When \(n=6\) (an RB15625 network), the second and fifth levels, which are expected to contain 1250 and 22 communities, turn out to be undetectable by our method. our method is to detect up to five community levels for the RB networks; to detect more community levels, an improved method that rescales the resolution ranges for different community levels more evenly, should be worth studying in the future. Figure 3: Communities in the RB125 network. (**a**–**d**): Four different community structures detected by our method within the RB125 network. In each subfigure, we exhibit different communities with different colors and shapes of nodes (note triangles in different directions also represent different communities).