# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### On the Hierarchical Ravas-Barabasi (RB) Networks
Following this regulation, on the \(m\)-th community level (\(2\leq m\leq n\), here \(m=1\) stands for the lowest level), it is each RB5\({}^{m+1}\) unit that splits into 4 \(m+2\) communities, thus, an RB5\({}^{m}\) network (\(n\geq m+1\)) will be divided into (4 \(m+2\)) \(\times\) 5\({}^{n-m-1}\) communities. Supplementary Figure S1a visualizes an RB625 network being divided into 14 communities on the third community level, and Table 1 summarizes the numbers of communities on different levels of the RB networks given by the above divisions. Next, we demonstrate how such divisions would be discovered by our method. We first try with \(\beta=1\), i.e., the original form of the community fitness function (Formula (1)). For all RB networks, with \(\beta=1\), we only detect two community levels: the highest (but trivial) level that merges the whole network into one single community, and the lowest level that divides the network into smallest communities that cannot split further; intermediate levels, if exist, are all missed. On the other hand, with \(\beta=1\), we detect different divisions for the lowest level. Except for our proposed divisions as listed in the first column of Table 1, we also detect the natural divisions, but only for two small RB networks: RB25 and RB125 (see Figures 2b and 3a); for larger RB networks, the natural divisions can no longer stand, since they violate the customary definitions of community too seriously. Moreover, with \(\beta=1\), an RB125 network can be divided into 26 communities--this can be done by breaking up only the central RB5 community of each RB125 unit (rather than each RB25 unit) into  
Figure 2: Communities in the RB25 network. (**a**) An RB5 network is a complete graph consisting of 5 nodes and 10 edges. We call node 0 the central node, and all other nodes peripheral nodes of the RB5 network. (**b**) An RB25 network is composed of five RB5 units, one in the center and four on the periphery. Every peripheral node of the peripheral unit is connected to the central node of the central unit, but different peripheral units are not connected to one another. Ideally, an RB25 network is expected to be divided into 5 communities: each RB5 unit makes a community. (**c**) A plausible revision to (**b**), which divides the RB25 network into 6 communities. Four peripheral RB5 units make four communities, and the central RB5 unit is divided into two communities: the central node makes one community and all peripheral nodes make the other. (**d**,**e**): Plateaus identified by our method with \(\beta=1\) and \(\beta=2\). The resolution parameter \(\alpha\) varies from 0 to \(2\beta-1\), with a stepwise increment \(\Delta\alpha=0.01\). At each resolution, we implement 1000 realizations of the Louvain algorithm, and identify the best-and-unique solutions and plateaus by the strategy described in Section 2.3. 2 communities, but preserving all other RB5 communities (as in Figure 3d). Compared to the divisions listed in Table 1, this alternative division can be understood as a result of a \"relaxed stringency\", with which an RB5\({}^{n}\) network (\(n\geq 3\)) can be divided into \(26\times 5^{n-3}\) communities; this explains the plateaus representing 26, 130, 650 and 3250 communities that emerge in Figure 3e and Supplementary Figures S1b and S2a,c. Since all these plateaus have similar community sizes (\(\leq\)5), we classify them all to the lowest community level. Moreover, with an even further relaxed stringency, where only the central RB5 community of each RB625 unit is broken up into 2 communities, an RB625 network can be divided into 126 communities. Yet such stringency has been over-relaxed so that it only produces a very tiny plateau for the RB625 network in Supplementary Figure S1b, but has not been observed anywhere else. The above results detected with \(\beta=1\) are still not satisfying; one major problem is that the highest and lowest community levels have occupied almost all the resolution scales; intermediate levels are seriously compressed and cannot be observed at all.