# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### Applications to Real-World Networks
As shown in Figure 6c,d, with varying resolution, our method detects 11 levels of communities with \(\beta=1\), and 10 levels with \(\beta=2\) in the karate club network; the numbers of communities include every value from 1 to 12. Surprisingly, these community levels are all stable and unique, whose topologies are exhibited in Supplementary Figure S6, forming roughly a hierarchical structure, with only minor \"reassembling\" of communities in the 5- and 6-community levels (detected with \(\beta=1\) only, denoted by the dashed rectangles in Supplementary Figure S6). Among all these levels, our 2-community division differs from the metadata--but it is fully consistent with the division suggested in [60,61], which has also been known as the only way to divide the karate club network into communities all defined in a strong sense [62]. Except for the 2-community level, communities of all the other levels can be properly combined to recreate the metadata division. For example, in Figure 6a,b, we exhibit our 4- and 5-community divisions, respectively, by nodes of different shapes. Obviously, combining circles and octagons in both Figure 6a,b roughly recovers [PERSON]'s club, while combing squares and hexagons (and also diamonds in Figure 6b) roughly recreates the officers'. Now, we compare our divisions with previous ones. The same as previous methods [13,59], our method also subdivides the metadata division into more communities. As shown in Figure 6a, our 4-community division is largely consistent with the division given by [PERSON] and [PERSON] through their shortest path betweenness method [13], except one node: node 10. We notice that node 10 has only two neighbors: node 3 joined [PERSON]'s club, and node 34 joined the officers'. It seems difficult to determine which choice for node 10 should be better than the other based on the network structure. On the other hand, compared to the metadata division, both our result and [PERSON]'s have misclassified node 9 to the officer's club, since node 9 evidently has more connections to the officers' club than to [PERSON]'s. Actually, in the original literature of [PERSON]'s [56], there are two metadata attributes recorded: the political leaning of each of the members and the faction they finally joined after the club fission. Previous literature on community detection only used the latter to evaluate the results, so that node 9 is almost always mislabeled. Considering the metadata on the political leaning of members (see Table 1 in [56]), node 9  
Figure 6: Communities detected in the karate club network. (**a**) Divisions given by our 4-community level and Newman–Girvan [13]. Our division is shown by nodes of different shapes (octagon, circle, square, and hexagon), while [PERSON] and [PERSON]’s is by nodes of different colors (yellow, red, blue and green). (**b**) Divisions given by our 5-community level and [PERSON] et al. [59]. Our division is shown by nodes of different shapes (octagon, circle, square, hexagon and diamond), while [PERSON] et al.’s is by nodes of different colors (yellow, red, blue and green). The dashed line drawn in both (**a,b**) divides the network into two parts, corresponding to a fission which had actually happened to the club. (**c,d**) Plateaus detected by our method from 1000 realizations of the Louvain algorithm at each resolution with \(\beta=1\) and \(\beta=2\). Numbers above the plateaus indicate the corresponding numbers of communities. was actually a weak supporter of the officer, but he chose to join [PERSON]'s club only for the convenience of a coming exam for his black belt. Meanwhile, node 10 was identified as a member of no faction, who may have probably chosen the faction randomly. As suggested by [55], the detected communities and the metadata may capture different aspects of the network structure, thus, some misclassification of nodes may also provide worthy information about the network. Alternatively, the simulated annealing approach applied by [PERSON] et al. results in a different division for the karate club network [59].