Fig. 1. Betweenness concept in networks and dynamical systems.

a A small network composed by nodes (colored circles) and links (black segments). The central node A behaves as a bottleneck for the network since all the paths connecting the blue with the red part must go through it. Indeed, node A, despite being attached to only two links, has the maximum value of betweenness centrality. b A set of Lagrangian trajectories (black lines) in a generic dynamical system. Each trajectory represents the evolution of an initial condition in space and time. Initial conditions, depending on the case studied, can be associated either to states of a system or to particles and agents moving across it. The yellow region A highlights a region where trajectories from different origins are choked into a narrow passage before being dispersed again. Similarly to node A in a, this region would qualitatively represent a bottleneck of the system and should be associated to a high value of betweenness.