Clustering coefficient and periodic orbits in flow networks

Rodriguez-Mendez, V.; Ser-Giacomi, E.; Hernandez-Garcia, E.
Chaos 27, 035803 (1-9) (2017)

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 tau 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 tau 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 tau reveals
the presence of periodic orbits of period 3tau 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.

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