The large-scale structural ingredients of the brain and neural connectomes have been identified in recent years. These are, similar to the features found in many other real networks: the arrangement of brain regions into modules and the presence of highly connected regions (hubs) forming rich-clubs. Here, we examine how modules and hubs shape the collective dynamics on networks and we find that both ingredients lead to the emergence of complex dynamics. Comparing the connectomes of C. elegans, cats, macaques and humans to surrogate networks in which either modules or hubs are destroyed, we find that functional complexity always decreases in the perturbed networks. A comparison between simulated and empirically obtained resting-state functional connectivity indicates that the human brain, at rest, lies in a dynamical state that reflects the largest complexity its anatomical connectome can host. Last, we generalise the topology of neural connectomes into a new hierarchical network model that successfully combines modular organisation with rich-club forming hubs. This is achieved by centralising the cross-modular connections through a preferential attachment rule. Our network model hosts more complex dynamics than other hierarchical models widely used as benchmarks.

This work has been supported by (G.Z.L.) the European Union Seventh Framework Programme FP7/2007–2013 under grant agreement number PIEF-GA-2012-331800, the German Federal Ministry of Education and Research (Bernstein Center II, grant no. 01GQ1001A), and the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 720270 (HBP SGA1). Y.C. and C.S.Z. were supported by the Hong Kong Baptist University (HKBU) Strategic Development Fund, the Hong Kong Research Grant Council (GRF12302914), HKBU FRG2/14-15/025 and the National Natural Science Foundation of China (No. 11275027). G.D. is supported by the European Research Council Advanced Grant: DYSTRUCTURE (295129) and by the Spanish Research Project PSI2013-42091-P; and M.L.K. by the European Research Council Consolidator Grant: CAREGIVING (615539).