The edges represent connections of the individuals outside the activities of the club. At some point, the administrator and the instructor of the club broke up due to a conflict between them. The club was separated into two groups supporting the administrator and the instructor. Figure 4 shows the network. Originally, there are two modules, www.selleckchem.com/products/PF-2341066.html which have 16 nodes (squares and pentagons in the figure) and 18 nodes (circles and triangles in the figure), respectively.Figure 4Zachary’s karate club network. Different shapes show the modules. M1: pentagon, M2: square, M3: triangle, M4: circle.We apply our proposed method to this network. The criterion (6) is satisfied until K = 4. The result is shown in Figure 4, with different shapes of the nodes denoting different modules.

The estimated connection probability matrix isP^=(0.3640.0730.0560.0360.0730.4800.0000.0000.0560.0000.2370.1080.0360.0000.1080.480).(11)From this matrix, it is easy to see that M3 and M4 are more likely to connect each other. With statistical tests, we can get that the connection probability among M3, M4, and M1 is the same. Although M2 has no connections to M3 and M4, it has a larger connection probability to M1 than M3, M4 to M1. Thus these four modules are on the same level. In [19], the authors considered constructing the hierarchical modular structure of this network too. At first, they also found four modules on the lowest level. Then they found that this network has two modules with some nodes (3, 9, 10, 14, 31) belonging to both of them. We did not consider the overlapping nodes in this article.

However, we can see that because these overlapping nodes belong to both M1 and M3, and they connect both parts closely, our method detect M1 and M3, M3 and M4 as having the same connectivity.3.3. Hierarchical Modular Structure in Yeast Gene Coexpression NetworkIn this section, we apply our proposed approach to analyze a gene coexpression network of yeast. The data set we use was generated by Brem and Kruglyak from a cross between two distinct isogenic strains BY and RM [23]. As described in [23], a total of 5740 ORFs were obtained after data preprocessing. In our analysis, we only use the 1,800 most differentially expressed genes as input to construct coexpression network and derive modules.

When constructing Carfilzomib the adjacency matrix of the network, we use the hard thresholding, that is: if the absolute value of Pearson correlation coefficient between two genes is greater than some given value, we assign an edge between them; otherwise, there is no edge. We compute the linear regression coefficient between the frequency of degree d (log 10(f(d))) and the log 10 transformed degree d (log 10(d)), and choose the threshold that leads to approximately scale free property of the network as described in [24]. Finally, the threshold is set to be 0.705, R^ is about 0.75.