Abstract
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 its 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.
Highlights
River networks have been a subject of research for many years
The flow paths in a river network can be described through a directed graph which can itself be represented by an N × N adjacency matrix A, where N is the number of nodes
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
Summary
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[1,2,3,4,5]. Consisting of branching channels, river networks’ display highly nonlinear dynamics and complex topology. 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[10,11,12,13]. To the best of our knowledge, the spectral properties of river network topology, such as eigenvalue distribution and spectral gap, have never been studied
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