Table 1.
Various definitions of global and local modularity. MΩ(s) and denote the number of links and the mean number of links in community s and the neighborhood of it, where
denotes the mean degree of network and NΩ(s) denotes the number of nodes in community s and the neighborhood of it.
Please refer to Methods section for γs.
Table 2.
Networks used in the experiments.
[A, B] denotes the parameter will vary from A to B. “1.5 kmax” denotes that cmax changes with kmax in the given proportion, while “0.015 N” denotes that the parameter varies with N in the given proportion. t1=2, t2=2, μ=0.2.
Fig 1.
Normalized mutual information (NMI) obtained by different modularity as a function of community-size difference (cmin-cmax) in the NET1 networks.
Parameters of networks: (a) N=1000, kmax=10; (b) N=1000, kmax=30; (c) N=5000, kmax=10; (d) N=5000, kmax=30 (see Table 1 for details of network parameters). Inset graphs show the fraction (fr) of affected nodes due to the merging of communities (i.e., the first-type resolution limit) by different methods as a function of community-size difference in the networks.
Fig 2.
NMI obtained by different modularity as a function of vertex-degree difference (kmax-km) in the NET2 networks.
Parameters of networks: (a) N=1000, cmax=150; (b) N=5000, cmax=150; (c) N=1000, cmax=1.5 kmax; (d) N=5000, cmax=1.5 kmax (see Table 1 for details of network parameters).
Fig 3.
NMI obtained by different modularity as network size in the NET3 networks.
Parameters of networks: (a) kmax=30, cmax=50; (b) kmax=30, cmax=150; (c) kmax=100, cmax=150; (d) kmax=0.015 N, cmax=0.020 N. (see Table 1 for details of network parameters).
Fig 4.
NMI of different methods as a function of γ in the NET4 networks with different heterogeneity of degree and community size (i.e., different values of kmax and cmax).
Parameters of networks: (a) kmax=30, cmax=150; (b) kmax=100, cmax=150; (c) kmax=30, cmax=600; (d) kmax=100, cmax=600. (see Table 1 for details of network parameters).
Fig 5.
Composite comparison of different methods in the LFR networks with different μ-values.
Parameters of networks: N=5000, km=10, kmax=100, cmin=10, and cmax=150 (see Table 1 for details of network parameters). “Mod” denotes the original Modularity. The optimal results is given for the modularity.
Fig 6.
Modularity obtained by different methods in real-world networks.
Fig 7.
Surprise obtained by different methods in real-world networks.
Fig 8.
Significance obtained by different methods in real-world networks.