River networks and optimal channel networks

Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Characterization of river drainage networks has been a subject of research for many years.

However, most previous studies have been limited to quantities which are loosely connected to the topological properties of these networks. In this work, through a graph-theoretic formulation of drainage river networks, we investigate the eigenvalue spectra of their adjacency matrix. First, we introduce a graph theory model for river networks and explore the properties of the network through 120 208 volt wiring diagram picture diagram base website adjacency matrix.

Next, we show that the eigenvalue spectra of such complex networks follow distinct patterns and exhibit striking features including a spectral gap in which no eigenvalue exists as well as a finite number of zero eigenvalues.

We show that such spectral features are closely related to the branching topology of the associated river networks. In this regard, we find an empirical relation for the spectral gap and nullity in terms of the energy dissipation exponent of the drainage networks. In addition, the eigenvalue distribution is found to follow a finite-width probability density function with certain skewness which is related to the drainage pattern.

Our results are based on optimal channel network simulations and validated through examples obtained from physical experiments on landscape evolution. These results suggest the potential of the spectral graph techniques in characterizing and modeling river networks. River networks have been a subject of research for many years. They are central to several processes occurring on river ecosystem and provide primary pathways to transport environmental fluxes such as water, nutrient, and sediment 12345.

Understanding and quantifying their structure and dynamics is essential for both advancing fundamental knowledge about their emergence and evolution as well as for management and prediction of environmental processes and fluxes operating upon them 123456789.

They have been shown to exhibit various properties such as self-similarity and scaling laws across a range of scales commonly observed in complex both natural and engineered networks 101112 Ranging from seminal works of Horton 1415 and Shreve 161718 which set the foundation of stream ordering schemes, several aspects related to river network geomorphology and topology have been explored using physical, theoretical, numerical and field approaches.

However, studies that specifically relate geometric and topologic properties of river network are still lacking. Along different lines, spectral graph theory has a long history 1920 and is a rapidly growing field in connection with complex networks Predominantly, spectral graph theory deals with the study of graphs through the eigenvalues and eigenvectors of their associated matrices.

river networks and optimal channel networks

Because of the generality of problems involving graphs, spectral graph techniques are deeply connected with different fields of science and engineering ranging from quantum chemistry 21 and communication networks 22 to computer science 23 and combinatorics 24 to mention a few. Considering the importance of the spectral graph techniques in all such areas and given recent advances in complex network characterization, of interest would be to explore the ramifications of these theories and techniques in one of the most interesting examples of naturally occurring complex networks; river drainage networks.

Note that graph theory has been previously used to study drainage network topology More recently, the topologic and dynamic complexity of delta channel networks have been investigated through a graph-theoretic approach 26 However, to the best of our knowledge, the spectral properties of river network topology, such as eigenvalue distribution and spectral gap, have never been studied.

Using a graph theoretic formulation, here we investigate the eigenvalue spectrum of the adjacency matrix of river networks. First, we consider drainage networks generated on two-dimensional lattices through an optimal channel network OCN model. We then discuss general properties of the adjacency matrix and the eigenvalue spectrum associated with such networks. The main characteristics of the eigenvalue spectrum are extracted and their relation with the topology of the river network is discussed.

The statistical behavior of the eigenvalues is also investigated and is compared with that of well-known networks. Finally, we explore examples from physical experiments on landscape evolution and show that our results are applicable to a variety of complex river networks formed under different external forcings. Adjacency matrix, A can be expressed as. Thus, the structure of a river network can be fully determined through the coordinates of its nodes and its adjacency matrix which respectively describe the geometry and the topology of the network.

In other words, a river network can be considered as a spanning tree on a two-dimensional regularly spaced square lattice grid of nodes N which can, in principle, be surrounded by an arbitrary shaped boundary describing the shape of a river basin Each node on this grid can only be connected to its eight nearest neighbors through a link.

The connecting links, are directed links representing the flow direction. Although each node can have an inflow from multiple upstream nodes, it can only have one outflow to the downstream node; thus each link is uniquely associated with its upstream node.In response to member requests for guidance and examples to help them effectively engage with members of their community to gather data and use it to address local, water-related problems, River Network is pleased to offer a new grant opportunity.

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Meet Your Network Meet the heroes safeguarding water for our families and communities. We Stand in Solidarity with the Black Community. Map: Who Protects Water?Lee Benda leebenda aol. Lee Benda, N. Hierarchical and branching river networks interact with dynamic watershed disturbances, such as fires, storms, and floods, to impose a spatial and temporal organization on the nonuniform distribution of riverine habitats, with consequences for biological diversity and productivity.

Abrupt changes in water and sediment flux occur at channel confluences in river networks and trigger changes in channel and floodplain morphology.

This observation, when taken in the context of a river network as a population of channels and their confluences, allows the development of testable predictions about how basin size, basin shape, drainage density, and network geometry interact to regulate the spatial distribution of physical diversity in channel and riparian attributes throughout a river basin.

The spatial structure of river networks also regulates how stochastic watershed disturbances influence the morphology and ages of fluvial features found at confluences.

Principles of fluvial geomorphology have guided the development of much of riverine ecology over the last half-century. A prominent example is the influential river continuum concept RCC; Vannote et al.

river networks and optimal channel networks

Based on early principles of fluvial geomorphology e. It predicts gradual adjustments of biota and ecosystem processes in rivers in accordance with the geomorphic perspective of gradual downstream changes in hydrologic and geomorphic properties. The linear perspective embodied in the RCC has dominated river ecology over the last 20 years Fisheralthough downstream interruptions in channel and valley morphology, caused by alternating canyons and floodplains, tributary confluences, and landslides, have long been observed.

Some have viewed these interruptions simply as adjustments to the original RCC e. In essence, river discontinuum perspectives highlight the nonuniform or patchy distribution of habitats and therefore emphasize habitat heterogeneity, expressed at the scale of meters to kilometers. Such heterogeneity also arises because of the human perception of scale, in which fluvial landforms are hierarchically organized from valley segments to stream bed particles Frissell et al.

Consequently, the idea of patchy and multiscale habitat formation and its related heterogeneity has imbued much current thinking in riverine ecology Frissell et al. Riverine ecology has also recognized the importance of physical disturbance e. Just as habitat patches create discontinuities in space, disturbances create discontinuities in time. Concepts emphasizing disturbance or watershed dynamics are generally applied in the context of a particular location within a watershed.

However, recent advances in understanding watershed disturbance regimes indicate how disturbance frequency and magnitude are organized by hierarchical and branching river networks Benda and Dunne abGomi et al. In sum, although the RCC's predictions of gradual downstream change in river attributes and associated biological processes are valid over orders of magnitude in river size, three other themes have arisen in riverine ecology over the past two decades in the effort to address how deviations arise from the expected mean state in physical attributes along a river profile.

These themes are 1 patchiness or heterogeneity, 2 stochastic disturbance, and 3 hierarchical scaling. This suite of concepts has been used to argue that riverine ecology should be guided by principles of landscape ecology, a discipline that incorporates a similar set of ideas Schlosser The purpose of this article is to develop a geomorphic framework in support of recent advances in river ecology.

To create this framework, which we call the network dynamics hypothesis, we developed testable predictions about how the spatial arrangement of tributaries in a river network interacts with stochastic watershed processes to influence spatiotemporal patterns of habitat heterogeneity. We begin with a general review of how tributary confluences modify channel morphology.

Then we describe how tributary confluence effects vary in terms of the specific attributes of a network's structure, including basin size, basin shape, network pattern, size difference between confluent channels, drainage density, confluence density, local network geometry, and the power law of stream sizes figure 1. Next, we describe how stochastic watershed disturbances such as floods, fire, and storms impose temporal heterogeneity on confluence effects, but in a predictable fashion that reflects the controls exerted by the underlying network structure.

Finally, we consider how the general principles developed in our hypothesis could advance the coupled disciplines of geomorphology and riverine biology.

By definition, a tributary is the smaller of two intersecting channels, and the larger is the main stem. Strictly speaking, a tributary junction, or confluence, is defined as the point where two different streams meet.

In the broader definition used in this article, a tributary junction is the valley floor environment influenced by tributaries and may include alluvial fans, terraces, secondary channels, and wider floodplains. The numerous bifurcations and confluences of distributaries in braided channel systems are not covered here. Three main types of processes are responsible for transporting sediment and organic material down tributaries to confluences with the main stem.Joinsubscribers and get a daily digest of news, geek trivia, and our feature articles.

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Launch the tool, locate the Channel header, and click it to sort by Wi-Fi channel. Here, we can see that channel 6 looks a bit cluttered — we might want to switch to channel 1 instead. To access it, hold the Option key and click the Wi-Fi icon on the menu bar at the top of your screen. Instead, you might as well just use the terminal.

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River landscapes and optimal channel networks

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Panel: Multi-Channel Networks

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river networks and optimal channel networks

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Contact Us.River networks represent a perfect example of a physical phenomenon that can be described by means of graph theory.

Channels and Valleys

Water collected by rainfall flows from one point to another one downstream in the river basin creates a spanning water flows uniformly on the terrain and therefore from every point of the basin we have water flow tree water cannot flow uphill.

Rivers on Earth and even those that might have been present on Mars all display similar statistical properties thereby calling for a model based on basic properties. A class of models named Optimal Channel Networks OCN derive the final configuration by minimising a given cost function. The physical inspiration for the minimization problem traces back to the ideas of Nobel laureate Prigogine on a general theory of irreversible processes in open dissipative systems.

Actually, theoretical results from OCN allowed to provide an explanation to universal allometric behaviour in a variety of different physical situations from species distribution to food webs optimisation alternative to the traditional approach. In the specific case of river networks, the OCN model postulates that the total gravitational energy loss in the system is minimised.

Empirical and theoretical works focus generally on two dimensional case, while recently inspired by vascular systems also the three dimensional case has been analysed. Here we devise some new analytical results that illustrate the role and the properties of the structure that minimises the cost function proposed in the ABM and we also provide some insight about the structure of the absolute minimum by varying some of the parameters of the model.

In what follows we will give a theoretical characterization of river networks and provide a simple rule to distinguish spanning trees from natural river trees. Furthermore, we extend the study of OCNs embedded on a lattice finding a lower and upper bound for the energy of an OCN in any dimension D. Location of Repository.

OAI identifier: oai:eprints. Suggested articles.Author contributions: P. Morris obtained the mathematical proofs; and P. Mastrandrea, R. Morris, and A. Reviewers: A. Optimal channel networks OCNs are a well-studied static model of river network structures.

We present exact results showing that every OCN is a natural river tree where a landscape exist such that the flow directions are always directed along its steepest descent.

Our results are significant in particular for applications where OCNs may be used to produce statistically identical replicas of realistic matrices for ecological interactions. We study tree structures termed optimal channel networks OCNs that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms.

Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent.

Results extend our capabilities in environmental statistical mechanics. River networks can be viewed as rooted trees that, when extracted from fluvial landscapes by topographic steepest descent directions, show deep similarities of their parts and the whole, often across several orders of magnitude, despite great diversities in their geology, exposed lithology, vegetation, and climate 1.

The large related body of observational data provides quintessential examples of real physical phenomena that can be effectively modeled by using graph theory 23. River networks are in fact the loopless patterns formed by fluvial erosion over a drainage basin. A specific class of spanning trees, called optimal channel networks OCNswas obtained by minimizing a specific functional 45later shown to be an exact property of the stationary solutions of the general equation describing landscape evolution 67.

The static properties and the dynamic origins of the scale-invariant structures of OCNs proved remarkable 1. OCNs are suboptimal that is, dynamically accessible given initial conditions and quenched randomness frustrating the optimum search configurations of a spanning network mimicking landscape evolution and network selection SI Appendix.

Empirical and theoretical works have generally focused on the two-dimensional 2D case, although recently inspired by vascular systemsthree-dimensional 3D settings have also been analyzed for an OCN embedded in a lattice 8. Several exact results have been derived for OCNs 6 — A model for a river network is obtained by taking a reasonably dense set of points on a terrain and then joining each point to a nearby point downhill. Experimental observations suggest that OCNs should satisfy the following properties: i each portion of the drainage basin has a single output; and ii a geomorphological stability condition holds.

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