Fig 1.
Probability distributions.
Fig 2.
Kp represents a clique with p vertices.
Table 1.
Numerical examples of modularity and Z-modularity for some ring of cliques networks.
Fig 3.
Network with two pairwise identical cliques.
Kp and Kq represent cliques with p and q vertices, respectively.
Table 2.
Numerical examples of modularity and Z-modularity for some networks with two pairwise identical cliques.
Fig 4.
Results for the planted l-partition model.
Each point is the result of averaging over 100 network realizations. The top and bottom bars represent the maximum and minimum values, respectively.
Fig 5.
Results for the LFR benchmark (n = 1000).
Each point is the result of averaging over 100 network realizations. The top and bottom bars represent the maximum and minimum values, respectively.
Fig 6.
Results for the LFR benchmark (n = 5000).
Each point is the result of averaging over 100 network realizations. The top and bottom bars represent the maximum and minimum values, respectively.
Fig 7.
Adjacency matrices for an LFR benchmark network.
(A) Ground-truth partition: 10 communities. (B) Optimal partition for Z-modularity: 81 communities and Inorm = 0.6942.
Fig 8.
Community structure for Hanoi graph H4.
(A) Optimal partition for Z-modularity: 27 communities, Z = 3.376, and Q = 0.6379. (B) Optimal partition for modularity: 9 communities, Z = 2.510, and Q = 0.7889.
Fig 9.
Community structure for Zachary’s karate club network: 6 communities, Z = 0.9266, Q = 0.3882, and Inorm = 0.4796.
Fig 10.
Community structure for Les Misérables network: 9 communities, Z = 1.490, and Q = 0.5245.
Fig 11.
Community structure for American college football network: 14 communities, Z = 2.111, Q = 0.5738, and Inorm = 0.9205.