Fig. 3: Detection of multiscale community structure with different heterogeneity.
a This network comprise four top-level communities (labeled as a, b, c, and d) with 400 nodes each and an inter-connection probability p1 = 0.0002, each of which further comprises four second-level communities with 100 nodes and p2 = 0.035 (e.g., community c comprises c1, c2, c3, and c4). The second-level communities are generated by the Barabási-Albert model45 with m = 7, which leads to an average degree 〈k〉 = 14. b shows the decision graph for the LS method when analyzing the network in (a). c displays the hierarchical structure formed by the local dominance between identified centers of each community. For better clarity, community centers are named by the community label instead of the real index of the node, and we only show the tree structure of these centers. The height difference indicates the li of the lower node. (d)-(f) is the same as (a)-(c), with only changing the generation process of second-level communities to the Erdős-Rényi random network with a connection probability p = 0.14, which still leads to the same average degree 〈k〉 = 14. In such a setting, similar to stochastic block models, nodes in the network are again relatively homogeneous. For better clarity, in (e) and (f) only top sixteen centers are labeled and their affiliation relation are visualized, and in total, LS detects 29 centers at the second-level for this network. For the multiscale network in (a), the LS method detects four top-level communities with F1 = 0.99 and 16 second-level communities with F1 = 0.56. For the network in d, the LS method detects four top-level communities with F1 = 0.89 and 29 second-level communities with F1 = 0.29. In both cases, the Louvain algorithm only obtain four communities, which corresponds to the first-level ones, with F1 equals 1, however, it cannot detect second-level partitions. By comparing results in (a)-(c) and in (d)-(f), we can find that our LS algorithm works well on networks with stronger heterogeneity. Results shown here correspond to just one realization, in multiple realizations, as every first- and second-level communities are equivalent, the label sequence in (b) and (e) and the tree structure in (c) and (f) may vary but have a consistent structure.
