# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 5 Conclusions  
Based on the community fitness function firstly proposed in [30], we made two improvements. First, we introduced a scaling factor \(\beta\) to amplify the varying range of the resolution parameter \(\alpha\), which also improves the scalability of the community fitness function. With this improvement, our method outperforms previous methods since it not only performs excellently on large heterogeneous networks without being affected by the resolution limit problem (Section 3.2), but also detects more multilevel communities, including the intermediate levels, in deep hierarchical networks (Section 3.1).  
The second improvement we made is that we suggested a strict definition for the term \"plateau,\" as well as a strict strategy to identify them. It has, on the one hand, avoided the ambiguous use of the term as in previous literature, and, on the other hand, remarkably improved the stability of our outputs. Our strategy automatically removes unstable results including the randomly detected communities as shown in Figure 11, selectively but inartificially. As a result, our outputs are very clean: all plateaus are stable, representing best and unique solutions obtained at each relevant resolution, without any redundancies or junk.  
Applied to real-world networks, our method discovers multilevel community structures, which are all stable and unique. Some of them have recovered known attributes, or metadata of the given network, and the rest are promising to inspire new insights into the network structure. Especially, we demonstrated that some different divisions suggested by previous literature actually reflect different aspects of the network structure under different resolution scales (Section 3.3).  
On extremely large networks, our method continues to detect communities effectively and efficiently. Its performance is at least equally as good as the widely accepted state-of-the-art algorithms such as modularity \(Q\), \(Q\gamma\) and Infomap. Comparing to other multiresolution methods, the explicit and finite varying range of our resolution parameter substantially facilitates our searching on any expected community structures suggested by the metadata (Section 3.4).  
Finally, our method can be implemented with any fast heuristic algorithm. In this paper, we carried it out with a classical Louvain algorithm, which turns out to be both effective and efficient in detecting communities for various types of complex networks.  
Our work in this paper has revealed the potential advantages of one class of \"scalable\" methods for community detection. Their outperformance on both large heterogeneous and deep hierarchical networks notwithstanding, for community detections in extremely deep or large networks, measures on the quality of communities that are specially developed (such as higher sensitivity, or the ability of zooming in on certain intermediate community levels), as well as more advanced heuristic optimization algorithms, are both worth pursuing in future studies.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13031774/s1](https://www.mdpi.com/article/10.3390/app13031774/s1)]([https://www.mdpi.com/article/10.3390/app13031774/s1](https://www.mdpi.com/article/10.3390/app13031774/s1)).  
Conceptualization, K.G. and X.R.; methodology, K.G.; software, K.G. and J.Z.; validation, K.G., X.R. and L.Z.; formal analysis, K.G. and X.R.; investigation, K.G.; writing--original draft preparation, K.G.; writing--review and editing, K.G., X.R. and L.Z.; funding acquisition, K.G. and L.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the \"One Thousand Talents Program\" of Sichuan Province with Grant No. 17 QR003, the Sichuan Science and Technology Program with Grant No. 2022 ZYD0013, and the Doctoral Research Fundings with Grant Nos. 16 az7112, KZ001212 and 21 zx7115, Southwest University of Science and Technology, Mianyang, Sichuan, P.R. China.