During the last few years, complex network approaches have demonstrated their great potentials as versatile tools for exploring the structural as well as dynamical properties of dynamical systems from a variety of different fields. Among others, recent successful examples include (i) functional (correlation) network approaches to infer hidden statistical interrelationships between macroscopic regions of the human brain or the Earth's climate system, (ii) Lagrangian flow networks allowing to trace dynamically relevant fluid-flow structures in atmosphere, ocean or, more general, the phase space of complex systems, and (iii) time series networks unveiling fundamental organization principles of dynamical systems. In this spirit, complex network approaches have proven useful for data-driven learning of dynamical processes (like those acting within and between sub-components of the Earth's climate system) that are hidden to other analysis techniques. This Focus Issue presents a collection of contributions addressing the description of flows and associated transport processes from the network point of view and its relationship to other approaches which deal with fluid transport and mixing and/or use complex network techniques.
The study of different types of flows is ubiquitous in the scientific literature. On the one hand, the mathematical description of dynamical systems is fundamentally based on the consideration of time-evolution operators that determine the system's dynamics in its associated phase space. On the other hand, the study of real-world flows has become an important field of study in fluid dynamics, with numerous geophysical (e.g., atmosphere and ocean), astrophysical, biological, and technological (e.g., traffic, infrastructures, and trade) applications.
Following the fundamental developments of the field in the 1990s, complex network theory1–3 has gained great attention across a variety of scientific disciplines and has become an integral part of the toolbox of nonlinear scientists. Applications of this concept include examples from many fields, ranging from physical to social systems and making use of a multitude of quantitative characteristics and algorithms.2,4–7 Important general achievements in this field include an improved understanding of the relationship between network structure and functionality, particularly in the context of synchronization processes.2,8
In the context of fluid dynamics, and more broadly in environmental modeling, complex network concepts have been recently used in a variety of situations which have in common that certain (sometimes small) spatial regions of a fluid, or of the Earth in geophysical applications, are represented by network nodes. Links between them, which complete the definition of the network, can represent two generic types of connectivity processes: statistical and material connections. In addition to regions of physical space, network nodes can also be identified with regions of the abstract phase space of a dynamical system producing some time series.
Defining network links by statistical connections means that two parts of the fluid have a link if the correlation (or another measure of statistical dependence) between two time series of some variables defined in them is sufficiently high. While the corresponding idea has been widely utilized in the neurosciences to establish information on the functional connectivity between different brain regions as inferred from multi-channel electroencephalogram (EEG) or functional magnetic resonance imaging (fMRI) measurements,9,10 an equivalent representation of spatio-temporal observations of climate variables has initiated the emerging field of climate network analysis.11–14 In an even broader context, this approach can be utilized as a nonlinear framework for the analysis of general multivariate time series (without any distinct spatial reference system), as exemplified by recent studies on economic networks created from financial returns of assets traded at stock markets or stock indices.15–17
Understanding climate dynamics as primarily reflecting mass transport in the atmosphere and ocean (i.e., the dynamics of fluid flows), functional climate networks based upon the statistical similarity between temporal variations of a climatic observable at different points in space (meteorological station data,18 grid points in global or regional reanalysis or climate model data sets,11 or even locations of paleoclimate archives19) serve as a prototypical example of complex network applications to flow systems based on a time-dependent spatial field of measurements. In almost all recent cases, scalar-valued properties like air temperature, pressure, precipitation, or related variables have been considered, while the explicit study of vector fields (in particular, wind fields) for this purpose is still in its infancy.20 However, it should be noted that from a fluid-dynamic perspective, networks generated from explicit observations of the velocity field of a flow system itself provide the most natural description of the spatio-temporal flow structure (and, thus, correspond to what one would refer to as a Eulerian flow network), whereas other variables like temperature provide useful statistical information but allow at most to indirectly infer information on the underlying flow dynamics.21,22
Despite this intrinsic limitation, correlation-based climate networks used for studying primarily atmospheric dynamics and also other components of the Earth system have recently provided important new results. For example, such network approaches have been successfully employed to identify precursors of abrupt transitions or tipping points in individual Earth system components. Specifically, the degree, assortativity, clustering coefficient, and degree-distribution kurtosis of correlation networks have found to rise when approaching a bifurcation point.23–25 In a similar spirit, percolation in such networks provides insights into approaching abrupt transitions.26 Temporal fluctuations in the linking patterns of surface air temperature networks allow tracing the state of large-scale climate phenomena like the El Niño Southern Oscillation (ENSO)12,27–29 and even distinguish between different types of El Niño and La Niña episodes.30,31 Beyond this application as a diagnostic tool, there have been first attempts towards establishing prognostic methods for anticipating the emergence of El Niño based on changes in the network properties.32,33
An alternative to defining links by statistical dependencies for establishing a complex network representation of a given flow system is to let the network links directly represent the amount of material transported among spatial locations,34–39 thus defining a Lagrangian flow network. If nodes represent regions of an abstract phase space, then the links characterize the evolution of the corresponding dynamical system from one (coarse-grained) state to another.34–36,40 In the context of Lagrangian approaches to fluid dynamics, a large body of work has been devoted to identifying almost-invariant or coherent sets in fluid flows from the associated transfer matrix, the so-called set-oriented approach to transport, by means of spectral methods.36,40–46 Identifying the transfer matrix with the adjacency matrix of a directed and weighted network allows in addition the use of modern community detection methods37,47,48 or path-finding algorithms.38,39
Finally, we emphasize that the transition matrix approach underlying the concept of Lagrangian flow networks has been employed in a more general context for studying sequences of discrete (possibly coarse-grained) states or dynamical patterns in arbitrary time series.49–51 Such transition networks belong to the increasing number of approaches for utilizing complex network methods for the analysis of general uni- or multivariate time series.52 Other notable examples of concepts for generating such time series networks include recurrence networks and related methods.53–55 We particularly note that the latter class of approaches (as well as bivariate generalizations thereof) has been used for the analysis of the dynamics of experimental flows based on point measurements of the flow velocity (see Gao et al.56 and references therein). In this context, time series networks of different types present yet another versatile toolkit for studying the dynamics of flow systems based on complex network methods.
The aim of this Focus Issue is to present a collection of papers enlarging the scope of using complex network theory for studying physical fluid flows as well as dynamical systems and presents some novel applications of complex network methods to both real-world and abstract (mathematical) flows. In what follows, we provide a brief overview on the different studies of this Focus Issue and put them into the context of the current state of research in this field. Finally, we briefly discuss some corresponding perspectives.
I. CORRELATION-BASED FLOW NETWORKS
A first group of papers in this Focus Issue analyzes the properties of networks constructed from statistical correlations between spatial locations:
Hlinka et al.57 assess the relevance of the bias associated with spurious detections of small-world proprieties in graphs constructed from time series inter-dependencies in dynamical systems. They present case study networks constructed from time series of both local brain activity and surface air temperatures. The understanding of such effects is key for interpreting the structural properties of spatially embedded networks in general58,59 (and, hence, also most types of flow networks), but in particular, it has direct implications for interpreting the results of functional network analysis in both climatic and neuro-physiological contexts.
Meng et al.60 study the percolation properties of a climate network constructed from lagged correlations of air temperatures and show that the percolation transition can be used as an early warning indicator of El Niño events. Their work contributes to the ongoing discussion on the spatial structure of El Niño's global teleconnectivity,59 i.e., the emergence of statistically significant relationships between climate variability at distinct parts of the globe.
Closely related to this problem is the study by Feng and Dijkstra,61 which investigates the stability of the slowly varying Pacific climate state that is key for understanding the emergence of ocean–atmosphere feedbacks that undermine the predictability of El Niño. In their work, they utilize the skewness of the degree distribution of a correlation network constructed from sea-surface temperatures (together with the recurrence rate and assortativity of the recurrence network representation of the closely related, scalar-valued NINO3.4 index) and demonstrate that these characteristics serve as easier-to-compute stability proxies than commonly used indices.
Molkenthin et al.62 analyze networks constructed from correlations among variables undergoing advection-diffusion-reaction dynamics. The network perspective is used to demonstrate that the local anisotropy (a member of a new class of spatial network characteristics) of edges incident to a given vertex provides useful information about the local geometry of the flow represented by the network. This opens a way to identify the underlying flow structures from observations of scalar variables at different points.
A second contribution by Hlinka et al.63 demonstrates the construction of directed climate networks based on linear Granger causality between globally distributed observations of surface air temperatures. They report that while the usual thresholding approach commonly employed in constructing climate networks yields a large number of network links masking the actually relevant processes, adopting instead a winner-takes-all approach reveals distinct spatial patterns of smooth information transport in the global temperature field which reflect the underlying dynamical structures of global wind fields.
While the former study63 presents one of the first attempts to combine causality concepts with spatially explicit climate network generation and analysis at global scale,64,65 Tirabassi et al.66 discuss how the even more elaborated concepts of renormalized partial directed coherence and directed partial correlation can be used in a climatological context. Both measures have recently proven their potentials in neuro-physiological signal analysis and are here for the first time employed to climate data in two specific case studies on ENSO–monsoon interactions as well as air–sea interactions in the South Atlantic Convergence Zone.
II. TRANSPORT-BASED FLOW NETWORKS
The following papers analyze networks representing fluid transport among different spatial locations and also the evolution of dynamical systems in phase space:
Lindner and Donner67 study tracer dynamics in a simple two-dimensional flow model and compute the spatial distribution of degree, eigenvalue centrality, and closeness, showing their relationship with the finite-time Lyapunov exponents. In particular, they demonstrate the potentials of path-based network measures with finite range, especially the so-called cutoff closeness, for highlighting transport barriers in the system.
In a similar fashion, Rodríguez-Méndez et al.68 employ Lagrangian flow networks to a hierarchy of paradigmatic models of flow systems. They specifically study the clustering coefficients of the resulting networks, revealing that they are able to approximate the locations of periodic trajectories in the associated phase space. By varying the time step τ as the fundamental parameter of their analysis setting, they introduce a kind of spectroscopic technique allowing to not only detect periodic orbits but also simultaneously identify their characteristic period.
The applicability of complex network approaches to Lagrangian trajectory data is further corroborated by the study of Banisch and Koltai.69 Here, the authors focus on the possibility to unravel coherent sets, i.e., flow regions that maintain their geometric integrity for relatively long times, from possibly sparse trajectory data. By transferring the idea of diffusion maps to the trajectory space, they demonstrate how to obtain dynamical coordinates that reveal the intrinsic low-dimensional organization of transport patterns within the flow.
Gelbrecht et al.70 propose a new method for constructing networks from atmospheric wind fields. Connecting nodes along a spatial environment based on the local wind flow, they derive a network representation of the low level atmospheric circulation that captures its most important characteristics.
Fujiwara et al.71 study the effect of small modifications of a flow on the spectral properties of Lagrangian flow networks, making use of perturbation-theoretic expansions inspired from corresponding methods in quantum mechanics. The specific types of perturbations considered include particle sources or sinks, as well as small modifications in the flow itself.
III. NETWORK ANALYSIS OF FLOW TIME SERIES
Finally, complex network methods are also useful in the analysis of temporal and spatio-temporal data series coming from fluid flow and dynamical systems:
Gao et al.72 analyze experimental impedance measurements from an oil-in-water bubbly flow. Modality transition-based networks employing the rank order among all pair-wise correlation coefficients from a four-channel signal are constructed from the ordering of correlations among the different impedance channels, and they are used to characterize different flow structures.
A second paper by Gao et al.73 uses a multivariate generalization of the recurrence network framework to study changes of oil-in-water flow behavior as the water-cut and flow velocity are modified. They demonstrate that the resulting changes in the average clustering coefficient and spectral radius of the thus constructed networks trace the observed phase transitions between different types of flow behavior.
McCullough et al.74 construct ordinal networks from discretely sampled chaotic time series generated by dynamical systems and regenerate new time series by taking random walks on the corresponding ordinal network. They investigate the extent to which the dynamics of the original time series are encoded in the ordinal networks and retained through the process of regenerating new time series by using several distinct quantitative approaches.
Last but not least, Kramer and Bollt75 demonstrate a network-based auto-synchronization method to estimate model parameters and unobserved variables from partially observed spatio-temporal data. The method is illustrated with a chaotic reaction-diffusion model describing ocean ecology.
IV. PERSPECTIVES
We expect that the papers included in this Focus Issue will stimulate further research at the interface between fluid dynamics, dynamical systems theory, and complex networks. From the current state of research as manifested in the different contributions to these collections, we emphasize a few key aspects where additional work appears desirable:
Data analysis vs. modeling: So far, many of the applications of complex network methods to studying flow systems have been mainly descriptive, i.e., used the different types of network construction approaches for inferring information on the characteristic flow structures based on observational or model data. It remains an open problem how to translate knowledge gained from such an empirical analysis of existing data into practical recommendations for improving existing models of real-world processes. While this might constitute a highly case-specific task, we emphasize that systematic data–model inter-comparison provides a general strategy for successive model improvement and that the properties of flow networks (constructed in whatever way) can assist this task by providing useful characteristics that can be used to identify and quantify existing discrepancies.
Diagnostic vs. prognostic applications: As mentioned above, most recent studies of climate networks and also other types of flow networks have mostly taken a diagnostic viewpoint. Counter-examples include studies addressing the changes in correlation network structures as the investigated system approaches a bifurcation point or other types of dynamical transition behavior. While this appears to be the most natural strategy for employing this type of flow network for prognostic purposes, Lagrangian flow networks appear specifically tailored for tracing changes in the transport properties of the underlying medium. As shown by Fujiwara et al.,71 the effect of minor changes to the flow structure or tracer distribution can be effectively understood by standard concepts of the perturbation theory.
Eulerian vs. Lagrangian viewpoints: To this end, only a few studies have addressed the relationship between correlation-based and trajectory-based network descriptions of the same system.21,22 Notably, network approaches directly utilizing the velocity field as a characteristic variable for network definition are still rare. Future work on this aspect will provide deeper insights into the spatial correlation structure underlying the flow variable itself (instead of some kind of secondary scalar-valued proxy), which might help establishing a clearer theoretical link between both viewpoints.
Beyond the aforementioned general challenges, this Focus Issue shall stimulate further applications of complex network methods to studying flow systems across disciplines. Quite a few of the studies published here have the potential of being transferred to other research questions or even disciplines. As guest editors of this Focus Issue, we are looking forward to the future development of this exiting multi-disciplinary topic.
ACKNOWLEDGMENTS
We are grateful to all authors of contributions to this Focus Issue, as well as the reviewers for their hard work in evaluating all manuscripts, which helped to shape this collection. The International Workshop “Complex Network Perspectives on Flow Systems” (CONFLOW), which was held on 21–22 September 2015 in Potsdam, Germany, and during which the idea of this Focus Issue was developed, has been financially supported by the European Commission via the Marie Curie ITN “LINC—Learning about Interacting Networks in Climate” (FP7-PEOPLE-2011-ITN) and the German Federal Ministry for Education and Research (BMBF) via the BMBF Young Investigator's Group CoSy-CC2 (“Complex Systems Approaches to Understanding Causes and Consequences of Past, Present and Future Climate Change” (Grant No. 01LN1306A). We acknowledge the logistic support of this workshop provided by Gabriele Pilz, Lyubov Tupikina, Catrin Ciemer, and Michael Lindner. Last but not least, we are grateful to the Chaos Editor-in-Chief, Jürgen Kurths, and the staff of the Editorial Office, who did a great job in bringing this Focus Issue into life.
