# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### Tests on Extremely Large Networks with Ground Truth Communities  
To test the performance of our method as the network's size increases, in this section we implement our approach to two very large real-world networks: the Amazon product co-purchasing network [67], and the DBLP collaboration network [67]. Table 2 shows the sizes, numbers of edges and numbers of communities contained in these networks. Both networks have metadata (i.e., \"ground truth\" communities) suggested in [67]. We observe these ground truth communities contain \"nesting\" structures, namely, some communities are proper subsets of other communities, implying that the ground truth communities are probably a mixture of communities detected at different resolutions. So, when we compare the ground truth communities to communities detected by other methods, we remove all the communities nested in other communities to avoid repeat counting. We choose the Amazon and DBLP networks for our test because they have different types of ground truths: the ground truth of the Amazon network contains relatively fewer communities, while the ground truth of the DBLP network contains much more communities. We select three state-of-the-art algorithms, including the standard modularity \(Q\) of Newman [27] and its multiresolution version \(Q_{7}\) generalized by [PERSON] [33] (see Formula (4)), as well as Infomap [37], to compare with our method proposed in this paper. We observe Infomap and \(Q\) perform differently: Infomap tends to split the network into more communities while \(Q\) tends to split it less, so we anticipate \(Q\) may work better for the Amazon network while Infomap would be more suitable for the DBLP network. Figures 9 and 10 exhibit the normalized mutual information (NMI, see Formula (5)) among the detected communities; here, we adopt the NMI as a statistical quality metric for our method comparison because on large networks it is not convenient to compare any detection results node by node. As expected, for the Amazon network, \(Q\) recovers the ground truth better than Infomap; the corresponding NMIs are 0.44 (\(Q\)) and 0.40 (Infomap); see Figure 9a. However, for the DBLP network, NMIs between the ground truth and the communities detected by \(Q\) and Infomap are, respectively, 0.37 and 0.69 (see Figure 10a), implying that Infomap performs better. Being a multiresolution approach, our method shows high consistencies to \(Q\) at low resolutions (i.e., with small values of \(\alpha\)), while performs more like Infomap at high resolutions. Since both the Amazon and the DBLP network are not multilayer, in this section we detect their communities with \(\beta\) = 1 (with \(\beta\) > 1 we obtain similar results). Figures 9b and 10b exhibit the NMIs between our communities and those detected by Infomap and \(Q\). The best fits between our method and \(Q\) emerge at \(\alpha\) = 0.03, NMI = 0.73 for the Amazon network, and \(\alpha\) = 0.05, NMI = 0.64 for the DBLP network. It should be noted that these NMIs already imply a high consistency between the compared methods, because when we check the results detected all by \(Q\) in multiple realizations for the same networks, the NMIs between two detection results with very close values of modularity (\(\Delta Q\) < 10\({}^{-3}\)) turn out to be 0.80 for the Amazon network and 0.63 for the DBLP network, which are close to the NMIs between our method and \(Q\). On the other hand, consist