# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### On the Hierarchical Ravas-Barabasi (RB) Networks
(**a**) shows a “natural” division for the network: each RB5 unit makes a community; (**b**) shows our proposed division in Table 1 on the lowest community level: each RB25 unit is divided into 6 communities as in Figure 2c, and the whole network is divided into 30 communities; (**c**) shows our proposed division on the second community level, which divides the network into 10 communities; (**d**) shows an alternative division of the network into 26 communities with a “relaxed stringency” on the lowest community level: only the RB5 unit in the center of the whole network is divided into 2 communities, while all other RB5 units are kept intact. (**e**,**f**): Plateaus obtained by our method with \(\beta=1\) and \(\beta=2\). In (**e**), with \(\beta=1\) only the first community level can be detected, but there emerge three different plateaus representing three different divisions for it: 30 communities (our proposed one as in (**b**)), 25 communities (the “natural” division as in (**a**)) and 26 communities (the “variant” as in (**d**)). In (**f**), with \(\beta=2\), all community levels are successfully detected: the first level exhibits only the 30-community division, and the second level only the 10-community division (as in (**b**,**c**), respectively); we believe these divisions are the most stable and “robust” among all potential divisions on the corresponding levels of the RB networks. Lastly, we show the performances of some earlier methods on the RB networks for comparison. The RB networks have two distinctive features: hierarchical and symmetrical. Methods without a tunable resolution parameter are not expected to detect multiple levels of communities from the RB networks, but they should be expected to perform well in recovering at least one level of community. In Supplementary Figure S4, we show the communities detected for an RB125 network by two well commended methods: (1) the standard modularity \(Q\) proposed by [PERSON] [48], optimized through a Louvain algorithm, and (2) Infomap [37]. We find both these methods inevitably produce divisions with randomness through breaking the symmetry of the network: all communities in Supplementary Figure S4 are detected in a random manner. For example, in Supplementary Figure S4a, the central RB25 unit is divided into three communities by randomly combing two of its four peripheral RB5 units with the central unit, and each of the peripheral RB25 units is divided into two communities by randomly choosing one of its peripheral RB5 units as an individual community--such a division could be the best in terms of modularity, but it is definitely not unique. In the context that network nodes are distinguishable, considering the symmetry of the network, we can easily calculate there exist 1536 divisions that are equivalent to Supplementary Figure S4a; within 1000 independent realizations, almost every single realization suggests a different division. Similar is the division given by Infomap in Supplementary Figure S4b, which has 256 equivalents. In contrast, community levels detected by our method, as in Table 1, retain both the hierarchy and the symmetry of the network. We emphasize that the divisions given by modularity \(Q\) and Infomap through breaking the network symmetry are not necessarily wrong; they can be useful in some scenarios. Yet our method is designed to selectively output the best-and-unique solutions; in our context, we believe the best-and-unique solutions are better correlated to the node attributes that we are interested in as in real-world networks. As for methods with tunable resolutions, we choose the multiresolution version of modularity \(Q\) generalized by [PERSON] and [PERSON] [33]; other methods as those in [5; 35] essentially perform identically [42].