# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### Applications to Real-World Networks  
Unlike synthetic networks, real-world networks have no implanted communities. Instead, there are observed discrete-valued node attributes, or metadata, being customarily used as a proxy of the ground truth [51, 55]. In early literature, it was common to validate the effectiveness of a community detection algorithm by its success in recovering the metadata: if the detected communities correlate with the metadata, then one can reasonably conclude that the corresponding algorithm is promising to work effectively in practice--but its opposite has been realized more recently as being not true; i.e., failing to fit the metadata does not necessarily signify the failure of the algorithm [55]. Since synthetic (i.e., artificial) networks may not be well representative of naturally occurring interactions, applications to real-world networks are still worth checking for community detection algorithms. In this section, we investigate three real-world networks: [PERSON]'s karate club network [56], [PERSON] et al. 's dolphins social network [57], and the American college football network [2]. We detect communities for these networks with our method, and compare our results with the metadata or \"standard divisions\" given by previous studies. Considering the recent viewpoint on metadata [55, 58], we do not intend to validate the effectiveness of our method by its performance on these real-world networks--or not merely that. Thus,  
Figure 5: Normalized mutual information (NMI) between the communities detected in larger and more heterogeneous LFR networks by our method and Infomap against the implanted communities with varying \(\mu\). The following network parameters are shared among all subfigures: average degree \(\langle k\rangle=20\), maximum degree \(k_{\max}=100\), power-law exponent of the degree distribution of nodes \(\tau_{1}=-2\), range of community sizes \(s\sim\) [10, 100]. Other parameters including the network size \(N\) and the power-law exponent of the community size distribution \(\tau_{2}\) are labelled in each subfigure: (**a**) \(N\) = 10,000, \(\tau_{2}=-2\); (**b**) \(N\) = 10,000, \(\tau_{2}=-3\); (**c**) \(N\) = 50,000, \(\tau_{2}=-2\); (**d**) \(N\) = 50,000, \(\tau_{2}=-3\). we do not manage to fit our results unconditionally to the metadata; instead, we try to put some insight into the differences between them. As \"the scientific value of a method is as much defined by the way it fails as by its ability to succeed\" [55], a different but reasonable outcome to the metadata can hopefully help us understand a different aspect of the network structure. Figure 6 shows the community structures within the karate club network. This network consists of 34 nodes representing 34 members of a karate club; connections between nodes imply consistent interactions between the corresponding members outside the club. Due to a disagreement between the club president ([PERSON]) and a part-time instructor ([PERSON]), the original club later split into two parts, the officers' club and [PERSON]'s; members of the original club also diverge to follow their own favorite leader (see the communities split by the dashed line in Figure 5(a), which are referred to by the \"metadata division\" in the next paragraphs). Community detection methods in previous literature attempted to recreate such a division by various models; among them the result given by [PERSON] and [PERSON] [13] is often taken as a \"standard\" division for the karate club network. Figure 6: \(Cont\). Applying our method to the karate club network, we find a distinctive difference between real-world networks and synthetic networks--that is, in terms of the number of communities, synthetic networks are implanted with discrete levels of communities, while real-world networks may display more \"continuous\" community levels.