Based on the above results, we construct the hierarchical structure in an agglomerative way (bottom-to-up). We directly use connection probability, which is computed selleck catalog from the clustering results through maximum likelihood estimation, to measure the distance between different modules. This connection probability matrix is denoted as P^0. First the maximum connection probability between different modules is found, and we assume it is P^i0,j00 with the corresponding two modules i0, j0 being recorded. The second largest connection probability for these two modules i0, j0 are also found, and we assume they are P^i0,k00 and P^j0,l00 with the corresponding modules being k0 and l0. To determine whether there is a hierarchical structure for these modules, we use Fisher exact test to see whether the connection probabilities P^i0,k00 and P^j0,l00 are the same as P^i0,j00.

That is, we need to test P^i0,j00=P^i0,k00 and P^i0,j00=P^j0,l00. Here we take a P value threshold to be 0.05. Three different cases may occur for these two relations. (1) Both of these two null hypotheses are rejected. In this case, there is hierarchical structure and the modules i0, j0 are on the lower level than k0 and l0. We combine the two modules i0 and j0 and take them as one module. (2) Only one of P^i0,j00=P^i0,k00 and P^i0,j00=P^j0,l00 is accepted. The corresponding modules having the same connection probability are combined together. We look for the next largest connection probability for these three modules and test the relationship again.

If two modules are tested to have the same connection probability, they are combined into one group, and the same step is implemented again. (3) Both of these two null hypotheses are accepted. These modules are taken as on the same level and combine together. We search the next largest connection probability to these four modules and do the statistical test until the hierarchical structure occurs or all the modules are combined together. After the above steps are finished, the connection probability between different modules is recalculated and recorded as P^1. The above search and test steps are repeated for P^1. Such steps are implemented recursively until all the modules are combined into one big module/network. For the statistical tests, we can also use t-test to test the relations between the connection probabilities if the distribution of the connections between different modules can be approximated by normal distribution.

With this Batimastat method, we can efficiently combine the modules with the same connection probability into the same level.3. Numerical Experiments In this section, we evaluate the performance of our proposed method through its application to several examples. We first start with two artificial networks having comparatively clear module structures. We then apply our method to two real networks to evaluate its performance.