Spectral Characterization of River Networks under Climate Change

River drainage networks are fascinating examples of self-organized complex networks in nature. In general, drainage network patterns arise as the result of a balance between various processes including hillslope diffusion, channel incision, and tectonics. Quantifying the role of these drivers on stream network structure has been a subject of research for many years. Most well-known measures, however, have been focused on geometrical and hydrological parameters which are loosely connected to the structural properties of the network. Here, we introduce an alternative measure for the topology of river networks using spectral graph methods. In doing so, we develop a graph-theoretic description of stream networks where streams and junctions are respectively represented by graph edges and nodes. The eigenvalue spectrum of the adjacency matrix associated with river network graphs is then obtained and investigated. We show that the eigenvalue spectrum exhibits interesting features that distinguish river networks from random graphs. In particular, the eigenvalue exhibits a finite spectral gap as well as a finite number of zeros which we refer to as nullity. We show that the spectral gap and nullity are strongly connected to the branching pattern of the network. In this study, we utilize computer-generated river networks produced through an optimal channel network (OCN) model which is based on the principle minimum energy expenditure. In addition, by manipulating rainfall in our model, we study the impact of non-uniform precipitation on reorganization of river networks and demonstrate that reorganization is reflected in the spectral measures of the river networks. These results illustrate the potential significance of spectral graph methods for exploring the effects of climate change on the evolution of river networks.