Clustering coefficient and periodic orbits in flow networks

Abstract

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 $3\tau$ 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.

11 pages, 11 figures. To appear in Chaos, April 2017

References in corpus (16)

• The structure and function of complex networks
• Temporal Networks
• Clustering in Complex Directed Networks
• Generalizations of the clustering coefficient to weighted complex networks
• Complex networks in climate dynamics - Comparing linear and nonlinear network construction methods
• Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves
• Transport in time-dependent dynamical systems: Finite-time coherent sets
• An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems
• Flow networks: A characterization of geophysical fluid transport
• Inferring long memory processes in the climate network via ordinal pattern analysis
• Most probable paths in temporal weighted networks: An application to ocean transport
• Correlation networks from flows. The case of forced and time-dependent advection-diffusion dynamics
• Optimally coherent sets in geophysical flows: A new approach to delimiting the stratospheric polar vortex
• Dominant transport pathways in an atmospheric blocking event
• Spatio-temporal organization of dynamics in a two-dimensional periodically driven vortex flow: a Lagrangian flow network perspective
• Edge anisotropy and the geometric perspective on flow networks

Cited by in corpus (8)

• Complex network approaches to nonlinear time series analysis
• Explicit and implicit network connectivity: Analytical formulation and application to transport processes
• Lagrangian Flow Network approach to an open flow model
• Network Measures of Mixing
• Crossroads of the mesoscale circulation
• Network analysis of Reynolds number scaling in wall-bounded Lagrangian mixing
• A perturbative approach to Lagrangian flow networks
• Characteristic signatures of Northern Hemisphere blocking events in a Lagrangian flow network representation of the atmospheric circulation