Hierarchical modularity in human brain functional networks

Abstract

The idea that complex systems have a hierarchical modular organization originated in the early 1960s and has recently attracted fresh support from quantitative studies of large scale, real-life networks. Here we investigate the hierarchical modular (or "modules-within-modules") decomposition of human brain functional networks, measured using functional magnetic resonance imaging in 18 healthy volunteers under no-task or resting conditions. We used a customized template to extract networks with more than 1800 regional nodes, and we applied a fast algorithm to identify nested modular structure at several hierarchical levels. We used mutual information, 0 < I < 1, to estimate the similarity of community structure of networks in different subjects, and to identify the individual network that is most representative of the group. Results show that human brain functional networks have a hierarchical modular organization with a fair degree of similarity between subjects, I = 0.63. The largest five modules at the highest level of the hierarchy were medial occipital, lateral occipital, central, parieto-frontal and fronto-temporal systems; occipital modules demonstrated less sub-modular organization than modules comprising regions of multimodal association cortex. Connector nodes and hubs, with a key role in inter-modular connectivity, were also concentrated in association cortical areas. We conclude that methods are available for hierarchical modular decomposition of large numbers of high resolution brain functional networks using computationally expedient algorithms. This could enable future investigations of Simon's original hypothesis that hierarchy or near-decomposability of physical symbol systems is a critical design feature for their fast adaptivity to changing environmental conditions.

Keywords: brain; graph theory; hierarchy; information; modularity; near-decomposability; network.

Figures

Figure 1

Methods. (A) Downsampled template. Starting from a binary version of the AAL template (left), the downsampling procedure will produce a template of small (64 voxels), equal size regions covering the original template (right). (B) Illustration of the Louvain method on a simple hierarchical graph. The algorithm starts by assigning a different module to each node (16 modules of single nodes). The method then consists of two phases that are repeated iteratively. The first phase is a greedy optimization (GO) where nodes adopt the community of one of their neighbours if this action results in an increase of modularity (typically, the community of the neighbour for which the increase is maximal is chosen). The second phase builds a meta-network (MN) whose nodes are the communities found in the first phase. We denote by “pass” a combination of these two phases. The passes are repeated until no improvement of modularity is possible and the optimal partition is found. When applied on this graph, the algorithm first finds a lowest non-trivial level made of four communities. The next level is the optimal level and is made of two communities.

Figure 2

Variability and similarity of brain functional network community structure between 18 different subjects. (A) Matrix showing the between-subject similarity measure for community structure at the highest level of the modular hierarchy. The pairwise similarity scores for the most representative subject are highlighted by a black rectangle. (B) Matrix showing the between-subject similarities for community structure at the lowest level of the modular hierarchy. (C) Scatter plot showing strong correlation of between-subject similarities at high and low levels of the modular hierarchy. Red points are similarities for the most representative subject.

Figure 3

Hierarchical modularity of a human brain functional network. (A) Cortical surface mapping of the community structure of the network at the highest hierarchical level of modularity, showing all modules that comprise more than 10 nodes. (B) Anatomical representation of the connectivity between nodes in colour-coded modules. The brain is viewed from the left side with the frontal cortex on the left of the panel and the occipital cortex on the right of the panel. Intra-modular edges are coloured differently for each module; inter-modular edges are drawn in black. (C) Sub-modular decomposition of the five largest modules (shown centrally) illustrates that the medial occipital module has no major sub-modules whereas the fronto-temporal modules has many sub-modules.

Figure 4

Topological roles of network nodes in intra- and inter-modular connectivity. (A) All nodes are plotted in the {P − z} plane of intra-modular degree z vs participation coefficient P; the solid lines partition the plane according to criteria for hubs vs non-hubs and connector, provincial, peripheral or kinless nodes. (B) Anatomical representation of the provincial hubs (circles), connector hubs (large squares) and connector nodes (small squares) of each of each of the five largest modules at the highest level of the modular hierarchy. (C) Topological representation of the network in using Fruchterman–Reingold algorithm (Fruchterman and Reingold, 1991) to highlight topological proximity of highly connected nodes; colour and shape of the nodes represent their modular assignment and topological role as above and in Figure 2.

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.