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Efficient Physical Embedding of Topologically Complex Information Processing Networks in Brains and Computer Circuits

Figure 2

Hierarchical modularity in nervous and computational systems.

Dendrograms displaying significant modular and sub-modular structure for (A) a very large scale integrated circuit, (B) the nematode worm C. elegans, (C) the human cortical anatomical network estimated using conventional MRI in 259 normal volunteers and (D) the human cortical anatomical network estimated using diffusion spectrum imaging (DSI) from an independent sample of 5 normal volunteers. The modularity, , of each of these matrices was estimated using the Louvain community detection algorithm [18]; 1-tailed t-tests were performed to determine where the modularity of the observed network was higher than the modularity of a functional random (p-value, ), and pure random (p-value, ) network. The matrices were decomposed into their sub-modules, and each sub-module was tested for modularity, , greater than functional and pure random networks (, ) of the same size as the module being tested. This process was iteratively performed: sub-modules were tested for non-random modularity, and if sub-sub-modules were identified in this way then each of them was in turn tested for non-random modularity. All modules shown in the decomposition had , except for those few indicated in gray () and blue (). Complete decompositions are shown for the VLSI and human brain MRI network; both the C. elegans and human brain DSI networks continue to deeper hierarchical levels, here not shown due to space constraints (see supplementary Text S1 for full decompositions). Insets The inset panels give a visual depiction of the hierarchical modularity of each system, which has been represented by a co-classification matrix where red/brown colors highlight modules or clusters of nodes with high local interconnectivity and relatively sparse connectivity to nodes in other modules [74]; see also Figure 1 for a schematic.

doi: https://doi.org/10.1371/journal.pcbi.1000748.g002