# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### Applications to Real-World Networks
In Figure 6b, we demonstrate that [PERSON] et al. 's division is fully consistent with our 5-community division, if we combine the smallest community (diamonds, i.e., nodes 27 and 30) with a larger community (squares). On this community level, node 10 is correctly classified (to the officer's club). Our result suggests that although [PERSON] and [PERSON] et al. 's divisions look different, they are probably results observed at different resolutions: they represent different community levels, and are basically both correct. Next, we move to the dolphins' social network (hereafter, we call it the \"dolphins' network\" for short). The dolphins' network was compiled by [PERSON] and his collaborators from seven years of filed studies on a bottlenose dolphin society living in Doubtful Sound, New Zealand [57; 63]. To our knowledge, the first version of this network was established in [63], including 40 individuals of dolphins. After that, an extended version including 62 nodes and 159 edges was published in [57], which is the \"dolphins' network\" widely studied by community detection literature, including this paper. Nodes of the network represent the population of dolphins, while edges reflect associations between dolphin pairs occurring more often than expected by chance [57; 64]. [PERSON] and [PERSON] firstly divided this network into two communities in [13], which allegedly correspond to a \"known division\" of the dolphins' society. However, as far as we know, such a \"known division\" was not included in the metadata recorded in [57; 63], thus, previous literature actually took [PERSON] and [PERSON]'s division as a standard division. The larger community of [PERSON]'s can be further divided into four smaller communities [13], as visualized in Figure 7a. Such divisions have been cited later by both [PERSON] and [PERSON] [64; 65]. In [64], the smallest community containing only two nodes (Zipfel and TSN83, i.e., the purple community at the top of Figure 7a) was merged into a larger community (the red community right in the middle of Group 1), so that the total number of communities decreased to four [64; 65]. Applying our method to the dolphins' network, we obtain similar results to the karate club network. Among the multiple community levels, our 2-community level exhibits the strongest plateau (except the trivial single-community level, see Figure 7c,d. The corresponding network division is visualized in Figure 7b by nodes of different shapes (rectangles and ellipses). Compared to the standard division given by [PERSON] and [PERSON], our 2-community division misclassifies only one node, SN89, among all 62 nodes, to a different group. We notice that SN89 has only two connections: one to SN100, the central node with the highest betweenness [64] in Group 1, and the other to Web, an individual in Group 2. It was said that the \"known division\" between the two groups of dolphins was due to a temporary leave of SN100: interactions between the two groups were restricted while SN100 was away and became more common when it reappeared [13; 64]. We argue that when SN100 was away, presumably the interaction between SN89 and Group 1 should also be cut off--however, its interaction with Group 2 can be maintained through the connection to Web. Therefore, on the 2-community level, our classification for SN89 should be more reasonable, which fits the ground truth better. However, shifting to a higher resolution, the result is different. In Figure 7b, we exhibit our 7-community division by nodes of different colors; the corresponding communities are also enclosed in different boxes for better visibility. It turns out that two of our smallest communities (yellow and green, on the bottom middle and upper right of Figure 7b) can be merged into the blue and red communities to perfectly recreate Newman and Girvan's 5-community division. On this level, node SN89 has also been reclassified to the \"right\" group.