Computing the rich club of the network and its comparison to the CCA

In order to compare the CCA model to the rich club (RC; ref. ¹⁰³), a global network was constructed by averaging individual networks across all subjects. A standard consensus threshold was applied after averaging by zeroing any connection not present in a simple majority of individuals (i.e., a connection was kept in the average matrix if the connection was available in more than 50% of the total number of subjects). Next, node degree was estimated on this averaged network using the brain connectivity toolbox⁹¹. The node-degree estimates were normalized by subtracting the minimum and dividing by the maximum. A threshold was then applied to the node degree to identify the RC. We used results from previous work¹⁰³ and set the threshold of 14% to separate RC from periphery. This means that the top 14% of the nodes with the highest normalized node degree were assigned to RC. More specifically, 54 of 376 regions in the network were assigned to the rich club and the remaining 322 to the periphery—see color-coded ball-stick representations in Fig. 4. The size of the nodes in Fig. 4 was scaled by the magnitude of the CCA loading of each node.

After identifying the RC and periphery, the association between RC/Periphery and the CCA was investigated. The spearman rank correlation was estimated between the node degree in each node and CCA loadings. A bootstrap test was used to estimate the mean, standard error, and p-values reported in “Results”. To visualize the correlations, a scatter plot comparing the CCA variable loadings and the participation coefficient of the RC is shown in Supplementary Fig. 4a. The points are color-coded for the rich club (blue) and periphery (gray). The ordinate of Supplementary Fig. 4a is the RC participation coefficient for each node. The participation coefficient was computed as the ratio of the number of connections from that node to other nodes in the same group (RC or periphery) to the number of connections from that node to the other groups¹⁰³. The full list of CCA loadings in relation to the participation coefficient was displayed (Supplementary Fig. 4b) to show that the majority of RC connections also have relatively high loadings within the CCA as well.