Clustering coefficient and periodic orbits in flow networks

We show that the clustering coefficient, a standard measure in network theory, when applied to flow networks, i.e., graph representations of fluid flows in which links between nodes represent fluid transport between spatial regions, identifies approximate locations of periodic trajectories in the flow system. This is true for steady flows and for periodic ones in which the time interval τ used to construct the network is the period of the flow or a multiple of it. In other situations, the clustering coefficient still identifies cyclic motion between regions of the fluid. Besides the fluid context, these ideas apply equally well to general dynamical systems. By varying the value of τ used to construct the network, a kind of spectroscopy can be performed so that the observation of high values of mean clustering at a value of τ reveals the presence of periodic orbits of period 3τ , which impact phase space significantly. These results are illustrated with examples of increasing complexity, namely, a steady and a periodically perturbed model two-dimensional fluid flow, the three-dimensional Lorenz system, and the turbulent surface flow obtained from a numerical model of circulation in the Mediterranean sea. The Lagrangian description of fluid dynamics, which focuses on the motion of the fluid particles as they are advected by the flow, provides a useful bridge between the theory of dynamical systems and the analysis of fluid transport and mixing, so that techniques and results can be transferred from one field to the other. Modern network theory has also been brought into contact with fluid dynamics and dynamical systems through the concept of flow networks, in which the motion of fluid particles between different regions is represented by links in a graph. In this paper, we use the flow network framework to show that the clustering coefficient, a standard measure in network theory, identifies periodic orbits, fundamental objects in the theory of dynamical systems, and is also of importance in the context of fluid motion.We acknowledge the financial support from Spanish Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional under grants LAOP, CTM2015-66407-P (MINECO/FEDER) and ESOTECOS projects FIS2015-63628-C2-1-R (MINECO/FEDER), FIS2015-63628-C2-2-R (MINECO/FEDER), and from the program “Investissements d'Avenir” launched by the French Government and implemented by ANR under grants ANR-10-LABX-54 MEMOLIFE and ANR-11-IDEX-0001-02 PSL* Research University.N