Functional rich clubs emerging from the diffusion geometry of complex networks

Real systems are characterized by complex patterns of interactions between their units, by dynamical processes on them, and by the interplay of the two. It is well-known that particular structures affect dynamical processes at different scales. Sometimes richly connected units are connected by costly, long-range links. In the brain, hubs form rich-clubs for integrating information between different brain regions, and many biological and social networks show this same structural organization. It remains, however, unclear if it alone enables a rapid communication between highly connected nodes or if a functional rich-club may emerge as a combination of direct links and longer paths between rich nodes. Here, we identify functional rich-clubs through the diffusion geometry, providing a new perspective to rich-club phenomena in complex networks. We show that weak structural rich-clubs may be functionally stronger, thanks to bridge nodes, while diffusion inside strong structural rich-clubs may be damped in modular networks.