Most probable paths in temporal weighted networks: An application to ocean transport

Coastal oceanographic connectivity at the global scale: a dataset of pairwise probabilities and travel times derived from biophysical modeling

Ocean currents are fundamental drivers of marine biodiversity distribution, mediating the exchange of genetic material and individuals between populations. Their effect ranges from creating barriers that foster isolation to facilitating long-distance dispersal, which is crucial for species expansion and resilience in the face of climate change. Despite the significance of oceanographic connectivity, comprehensive global estimates remain elusive, hindering our understanding of species' dispersal ecology and limiting the development of effective conservation strategies. We present the first dataset of connectivity estimates (including probability of connectivity and travel time) along the world's coastlines. The dataset is derived from Lagrangian simulations of passive dispersal driven by 21 years of ocean current data and can be combined with species' biological traits, including seasonality and duration of planktonic dispersal stages. Alongside, we provide coastalNet, an R package designed to streamline access, analysis, and visualization of connectivity estimates. The dataset provides a new benchmark for research in oceanographic connectivity, enabling a deeper exploration of the complex dynamics of coastal marine ecosystems and informing more effective conservation strategies.

Comparing temporal and aggregated network descriptions of fluid transport in the Mediterranean Sea

Ocean currents exhibit strong time dependence at all scales that influences physical and biochemical dynamics. Network approaches to fluid transport permit to address explicitly how connectivity across the seascape is affected by the spatiotemporal variability of currents. However, such temporal aspect is mostly neglected, relying on a static representation of the flow. We here investigate the role of current variability on networks describing physical transport across the Mediterranean basin. We first focus on degree distributions and community structure comparing ensembles of temporal networks that explicitly resolve time dependence and their aggregated, i.e., time-averaged, counterparts. Furthermore, we explore the implications of the two approaches in a simple reaction dispersal model for a generic tracer. Our analysis evidences that aggregation induces structural network changes that cannot be easily avoided, not even introducing a pruning of the aggregated adjacency matrix. We also highlight that, depending on the time scales considered, the importance of the temporal features of the networks can vary significantly. Finally, we find that the tracer evolution obtained from a temporal dispersal kernel cannot be always approximated by aggregated adjacency matrices, in particular during transients of the dynamics.

A new measure of node centrality on schedule-based space-time networks for the designation of spread potential

Node centrality is one of the most frequently revisited network theoretical concepts, which got many calculation method alternatives, each of them being conceived on different empirical or theoretical network abstractions. The vast majority of centrality measures produced up to date were conceived on static network abstractions (the so-called "snapshot" networks), which arguably are less realistic than dynamic (temporal) network abstractions. The new, temporal node centrality measure that we offer with this article, is based on an uncommon abstraction, of a space-time network derived from service schedules (timetables). The proposed measure was designed to rank nodes of a space-time network based on their spread or transmission potential, and was subsequently implemented on the network of sea ferry transportation derived from the aggregated schedules for sea ferry liner shipping services in Europe, as they occurred in the month of August, 2015. The main feature of our measure, named "the Spread Potential", is the evaluation of the potential of a node in the network for transmitting disease, information (e.g. rumours or false news), as well as other phenomena, whichever support a space-time network abstraction from regular and scheduled services with some known carrying capacities. Such abstractions are, for instance, of the transportation networks (e.g. of airline or maritime shipping or the wider logistics (delivery) networks), networks of medical (hospital) services, educational (teaching) services, and virtually, of any other scheduled networked phenomenon. The article also offers the perspectives of the measure's applicability on the non-scheduled space-time network abstractions.

Spatial coalescent connectivity through multi-generation dispersal modelling predicts gene flow across marine phyla

Gene flow governs the contemporary spatial structure and dynamic of populations as well as their long-term evolution. For species that disperse using atmospheric or oceanic flows, biophysical models allow predicting the migratory component of gene flow, which facilitates the interpretation of broad-scale spatial structure inferred from observed allele frequencies among populations. However, frequent mismatches between dispersal estimates and observed genetic diversity prevent an operational synthesis for eco-evolutionary projections. Here we use an extensive compilation of 58 population genetic studies of 47 phylogenetically divergent marine sedentary species over the Mediterranean basin to assess how genetic differentiation is predicted by Isolation-By-Distance, single-generation dispersal and multi-generation dispersal models. Unlike previous approaches, the latter unveil explicit parents-to-offspring links (filial connectivity) and implicit links among siblings from a common ancestor (coalescent connectivity). We find that almost 70 % of observed variance in genetic differentiation is explained by coalescent connectivity over multiple generations, significantly outperforming other models. Our results offer great promises to untangle the eco-evolutionary forces that shape sedentary population structure and to anticipate climate-driven redistributions, altogether improving spatial conservation planning.

Network and geometric characterization of three-dimensional fluid transport between two layers

We consider transport in a fluid flow of arbitrary complexity but with a dominant flow direction. One of the situations in which this occurs is when describing by an effective flow the dynamics of sufficiently small particles immersed in a turbulent fluid and vertically sinking because of their weight. We develop a formalism characterizing the dynamics of particles released from one layer of fluid and arriving in a second one after traveling along the dominant direction. The main ingredient in our study is the definition of a two-layer map that describes the Lagrangian transport between both layers. We combine geometric approaches and probabilistic network descriptions to analyze the two-layer map. From the geometric point of view, we express the properties of lines, surfaces, and densities transported by the flow in terms of singular values related to Lyapunov exponents, and define a specific quantifier, the finite depth Lyapunov exponent. Within the network approach, degrees and an entropy are considered to characterize transport. We also provide relationships between both methodologies. The formalism is illustrated with numerical results for a modification of the ABC flow, a model commonly studied to characterize three-dimensional chaotic advection.

Characteristic signatures of Northern Hemisphere blocking events in a Lagrangian flow network representation of the atmospheric circulation

In the past few decades, boreal summers have been characterized by an increasing number of extreme weather events in the Northern Hemisphere extratropics, including persistent heat waves, droughts and heavy rainfall events with significant social, economic, and environmental impacts. Many of these events have been associated with the presence of anomalous large-scale atmospheric circulation patterns, in particular, persistent blocking situations, i.e., nearly stationary spatial patterns of air pressure. To contribute to a better understanding of the emergence and dynamical properties of such situations, we construct complex networks representing the atmospheric circulation based on Lagrangian trajectory data of passive tracers advected within the atmospheric flow. For these Lagrangian flow networks, we study the spatial patterns of selected node properties prior to, during, and after different atmospheric blocking events in Northern Hemisphere summer. We highlight the specific network characteristics associated with the sequence of strong blocking episodes over Europe during summer 2010 as an illustrative example. Our results demonstrate the ability of the node degree, entropy, and harmonic closeness centrality based on outgoing links to trace important spatiotemporal characteristics of atmospheric blocking events. In particular, all three measures capture the effective separation of the stationary pressure cell forming the blocking high from the normal westerly flow and the deviation of the main atmospheric currents around it. Our results suggest the utility of further exploiting the Lagrangian flow network approach to atmospheric circulation in future targeted diagnostic and prognostic studies.

Lagrangian betweenness as a measure of bottlenecks in dynamical systems with oceanographic examples

The study of connectivity patterns in networks has brought novel insights across diverse fields ranging from neurosciences to epidemic spreading or climate. In this context, betweenness centrality has demonstrated to be a very effective measure to identify nodes that act as focus of congestion, or bottlenecks, in the network. However, there is not a way to define betweenness outside the network framework. By analytically linking dynamical systems and network theory, we provide a trajectory-based formulation of betweenness, called Lagrangian betweenness, as a function of Lyapunov exponents. This extends the concept of betweenness beyond the context of network theory relating hyperbolic points and heteroclinic connections in any dynamical system to the structural bottlenecks of the network associated with it. Using modeled and observational velocity fields, we show that such bottlenecks are present and surprisingly persistent in the oceanic circulation across different spatio-temporal scales and we illustrate the role of these areas in driving fluid transport over vast oceanic regions. Analyzing plankton abundance data from the Kuroshio region of the Pacific Ocean, we find significant spatial correlations between measures of diversity and betweenness, suggesting promise for ecological applications.

Explicit and implicit network connectivity: Analytical formulation and application to transport processes

Connectivity is a fundamental structural feature of a network that determines the outcome of any dynamics that happens on top of it. However, an analytical approach to obtain connection probabilities between nodes associated with to paths of different lengths is still missing. Here, we derive exact expressions for random-walk connectivity probabilities across any range of numbers of steps in a generic temporal, directed, and weighted network. This allows characterizing explicit connectivity realized by causal paths as well as implicit connectivity related to motifs of three nodes and two links called here pitchforks. We directly link such probabilities to the processes of tagging and sampling any quantity exchanged across the network, hence providing a natural framework to assess transport dynamics. Finally, we apply our theoretical framework to study ocean transport features in the Mediterranean Sea. We find that relevant transport structures, such as fluid barriers and corridors, can generate contrasting and counterintuitive connectivity patterns bringing novel insights into how ocean currents drive seascape connectivity.

Centrality metric for the vulnerability of urban networks to toxic releases

The dispersion of airborne pollutants in the urban atmosphere is a complex, canopy-driven process. The intricate structure of the city, the high number of potential sources, and the large spatial domain make it difficult to predict dispersion patterns, to simulate a great number of scenarios, and to identify the high-impact emission areas. Here we show that these complex transport dynamics can be efficiently characterized by adopting a complex network approach. The urban canopy layer is represented as a complex network. Street canyons and their intersections shape the spatial structure of the network. The direction and the transport capacity of the flow in the streets define the direction and the weight of the links. Within this perspective, pollutant contamination from a source is modeled as a spreading process on a network, and the most dangerous areas in a city are identified as the best spreading nodes. To this aim, we derive a centrality metric tailored to mass transport in flow networks. By means of the proposed approach, vulnerability maps of cities are rapidly depicted, revealing the nontrivial relation between urban topology, transport capacity of the street canyons, and forcing of the external wind. The network formalism provides promising insight in the comprehensive analysis of the fragility of cities to air pollution.

Markov-chain-inspired search for MH370

Markov-chain models are constructed for the probabilistic description of the drift of marine debris from Malaysian Airlines flight MH370. En route from Kuala Lumpur to Beijing, MH370 mysteriously disappeared in the southeastern Indian Ocean on 8 March 2014, somewhere along the arc of the 7th ping ring around the Inmarsat-3F1 satellite position when the airplane lost contact. The models are obtained by discretizing the motion of undrogued satellite-tracked surface drifting buoys from the global historical data bank. A spectral analysis, Bayesian estimation, and the computation of most probable paths between the Inmarsat arc and confirmed airplane debris beaching sites are shown to constrain the crash site, near 25^°S on the Inmarsat arc.

From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data

One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time-and-space semidistance that comes from the "best" approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, and hence they occur as extremal regions on a spanning structure of the state space under this semidistance-in fact, under any distance measure arising from the physical notion of transport. Based on this notion, we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.

Spatio-temporal organization of dynamics in a two-dimensional periodically driven vortex flow: A Lagrangian flow network perspective

We study the Lagrangian dynamics of passive tracers in a simple model of a driven two-dimensional vortex resembling real-world geophysical flow patterns. Using a discrete approximation of the system's transfer operator, we construct a directed network that describes the exchange of mass between distinct regions of the flow domain. By studying different measures characterizing flow network connectivity at different time-scales, we are able to identify the location of dynamically invariant structures and regions of maximum dispersion. Specifically, our approach allows us to delimit co-existing flow regimes with different dynamics. To validate our findings, we compare several network characteristics to the well-established finite-time Lyapunov exponents and apply a receiver operating characteristic analysis to identify network measures that are particularly useful for unveiling the skeleton of Lagrangian chaos.

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.

Directed weighted network structure analysis of complex impedance measurements for characterizing oil-in-water bubbly flow

Characterizing the flow structure underlying the evolution of oil-in-water bubbly flow remains a contemporary challenge of great interests and complexity. In particular, the oil droplets dispersing in a water continuum with diverse size make the study of oil-in-water bubbly flow really difficult. To study this issue, we first design a novel complex impedance sensor and systematically conduct vertical oil-water flow experiments. Based on the multivariate complex impedance measurements, we define modalities associated with the spatial transient flow structures and construct modality transition-based network for each flow condition to study the evolution of flow structures. In order to reveal the unique flow structures underlying the oil-in-water bubbly flow, we filter the inferred modality transition-based network by removing the edges with small weight and resulting isolated nodes. Then, the weighted clustering coefficient entropy and weighted average path length are employed for quantitatively assessing the original network and filtered network. The differences in network measures enable to efficiently characterize the evolution of the oil-in-water bubbly flow structures.

Introduction to Focus Issue: Complex network perspectives on flow systems

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.

Correlation Networks from Flows. The Case of Forced and Time-Dependent Advection-Diffusion Dynamics

Complex network theory provides an elegant and powerful framework to statistically investigate different types of systems such as society, brain or the structure of local and long-range dynamical interrelationships in the climate system. Network links in climate networks typically imply information, mass or energy exchange. However, the specific connection between oceanic or atmospheric flows and the climate network's structure is still unclear. We propose a theoretical approach for verifying relations between the correlation matrix and the climate network measures, generalizing previous studies and overcoming the restriction to stationary flows. Our methods are developed for correlations of a scalar quantity (temperature, for example) which satisfies an advection-diffusion dynamics in the presence of forcing and dissipation. Our approach reveals that correlation networks are not sensitive to steady sources and sinks and the profound impact of the signal decay rate on the network topology. We illustrate our results with calculations of degree and clustering for a meandering flow resembling a geophysical ocean jet.

Dominant transport pathways in an atmospheric blocking event

A Lagrangian flow network is constructed for the atmospheric blocking of Eastern Europe and Western Russia in summer 2010. We compute the most probable paths followed by fluid particles, which reveal the Omega-block skeleton of the event. A hierarchy of sets of highly probable paths is introduced to describe transport pathways when the most probable path alone is not representative enough. These sets of paths have the shape of narrow coherent tubes flowing close to the most probable one. Thus, even when the most probable path is not very significant in terms of its probability, it still identifies the geometry of the transport pathways.