1. Artificial Networks 3.1.1. A Network Composed of Cliques We consider a network with 200 nodes, which is composed of 4 cliques. The sizes of the cliques are 90, 30, 40, and 40. The connections between different cliques are randomly generated with Imatinib Mesylate structure the following probability:P=(1.0000.2000.0020.0030.2001.0000.0050.0100.0020.0051.0000.0300.0030.0100.0301.000).(8)The pattern of the adjacency matrix is shown in Figure 2(a). From upper-left to lower-right, we denote the four modules as M1, M2, M3, and M4, which correspond to the position in the connection probability matrix. We can see the hierarchical structure of the network from the adjacency matrix. We apply our proposed method to this network. The condition (6) is satisfied until K = 4. The estimated connection probability matrix isP^=(1.

0000.2050.0030.0030.2051.0000.0060.0090.0030.0061.0000.0290.0030.0090.0291.000).(9)Figure 2Example of hierarchical modular network structure. (a) Pattern of the adjacency matrix; (b) the hierarchical structure of the network. We apply statistical tests to the corresponding modules, and finally we get the hierarchical structure as shown in Figure 2(b). The values on the hierarchical tree is the estimated connection probability of the corresponding modules. On the lowest level, there are four modules. If the tree is cut between 0.205 and 0.029, there are three modules while if the cutoff is greater than 0.029, there are only two modules. These results are consistent with the network generation strategy.3.1.2. A Randomly Generated Network In this example, we also consider a network with 200 nodes and 4 modules.

The size of each module is 10, 45, 45, and 100. We set the degree of each node within its module to be 6, 15, 15, and 30. Then the connections between different nodes are randomly generated. We keep all the edges generated for each node. So finally the average degree within each module is greater than the prespecified number. The connection probability between different modules is 0.002. The pattern of the adjacency matrix is shown in Figure 3. From upper-left to lower-right, the four modules are M1, M2, M3, and M4, respectively. With our proposed method, the network is partitioned into four modules correctly on the lowest level and the estimated connection probability isP^=(0.2980.0020.0020.0030.0020.3280.0020.0040.0020.0020.3210.0000.0030.0040.0000.560).

(10)By using the statistical tests, these four modules are determined as parallel modules, which is the same as that in our network generation strategy.Figure 3Pattern of the adjacency matrix for the randomly generated network. 3.2. Karate Club Network We consider the Brefeldin_A Zachary’s network of karate club members [22] in this example. There are 34 nodes in this network corresponding to the members in a karate club.