Figure 5.

Methodological issues in analysis of hierarchical modularity. (A) Validation of the Louvain method for hierarchical decomposition on a benchmark network defined in Sales-Pardo et al. (2007). The network is naturally made of 64, 16 and 4 modules of 10, 40 and 160 nodes respectively. The separability of different levels of the benchmark network is controlled by the parameter ρ. We calculate the normalized information between the lowest non-trivial level partition and the natural partition of 64 modules (solid curve), and between the second level partition and the natural partition of 16 modules (dashed curve). After averaging over 20 different realizations of the network, our simulations show an excellent agreement as mutual information is above 0.95 for values of ρ up to 1.5 for the lowest non-trivial and intermediate levels. (B) Modularity values at the highest and lowest levels of hierarchical community structure in a representative brain network (Subject ID 2, in red) and for networks obtained from 100 randomizations of the original time-series (in green), and for networks obtained by 100 randomizations of the original adjacency matrix. (C) Similarity measures between highest level partitions (left) and non-trivial lowest level partitions (right) obtained by thresholding the original network to retain different number of highest correlations as edges.