The diverse club

Abstract

A complex system can be represented and analyzed as a network, where nodes represent the units of the network and edges represent connections between those units. For example, a brain network represents neurons as nodes and axons between neurons as edges. In many networks, some nodes have a disproportionately high number of edges as well as many edges between each other and are referred to as the "rich club". In many different networks, the nodes of this club are assumed to support global network integration. Here we show that another set of nodes, which have edges diversely distributed across the network, form a "diverse club". The diverse club exhibits, to a greater extent than the rich club, properties consistent with an integrative network function-these nodes are more highly interconnected and their edges are more critical for efficient global integration. Finally, these two clubs potentially evolved via distinct selection pressures.

Fig. 1

Topology of the diverse and rich clubs in the human and C. elegans. a Visualization of a single C. elegans functional network, labeled according to the community affiliation detected by the Infomap algorithm. b, c The C. elegans’ diverse club and rich club. Nodes in red represent the maximum value for the given metric (participation coefficient or strength), yellow is median, and blue is the minimum. Edges are colored by the mix between the two nodes each edge connects. d Visualization of the human resting-state network labeled according to the community affiliation using the InfoMap algorithm. e, f The human diverse club and the rich club. In both networks, the diverse club clusters in the center of the layout, while the rich club forms clusters on the periphery. g The rich club and the diverse club (mean participation coefficient and strength across tasks and densities), along with nodes that are members of both clubs, are shown on the cortical surface of the human brain. h The mean percentage of each human cognitive system that is comprised of nodes from each club (analyzing all densities and tasks). For ease of interpretation, a canonical division of nodes into cognitive systems and names are used in h, while all other analyses and figures use the community detection calculated here

Fig. 2

Clubness for the rich and diverse clubs in every network. a–f In each network, the mean across network densities (human, functional C. elegans) or community detection runs (macaque, structural C. elegans, US power grid, flight traffic) for the clubness is plotted, with 95% confidence intervals shaded. In every network, as the rank increased and only nodes with a high participation coefficient (blue) or strength (green) are included in the club, the diverse club is typically higher in clubness than that of the rich club

Fig. 3

Distribution of nodes in clubs and communities. a The percentage of nodes in human networks that are in both the rich and diverse clubs. Zero represents that no nodes were members of both clubs, and 100% represents that the clubs are identical. b, c The percentage of communities in the network that contain a rich club node b or a diverse club node c. For each task, we calculate this for each density, utilizing the community detection results from that density. Thus, error bars represent the distribution of results across different densities

Fig. 4

A generative modular and efficient model exhibits a diverse club. a–f Six features of the network were analyzed across different ratios of maximizing Q (modularity, see Methods section, Eqs. 1–4) and E (efficiency, inverse of the sum of shortest paths between all nodes, see Methods section, Eq. 6). We used the Kullback–Leibler divergence to measure the degree and participation coefficient fits; thus, lower values indicate higher similarity (see Methods section, Eq. 9). 100 models were run at each ratio in 0.01 steps. Each value’s mean and 95% confidence intervals (shaded) are shown. g At ratios of 0.7–0.8 between weighting Q and weighting E, a balance between these six variables was achieved. h The average clubness across 1000 iterations for each rank for the diverse and rich clubs in the generative model at a ratio of 0.75 and the random model, as well as the t-test at each rank between the clubness of the diverse club in the model and the clubness of the diverse club in the random model (similar results from ratios of 0.70 and 0.80 are shown in Supplementary Fig. 66). Only the diverse club in the model has a high clubness