Controllability and heterogeneity of river networks using spectral graph theory approach

River networks are important landscape features that have been extensively studied over many years. While seminal works have focused on characterizing the topological properties of river networks, the quantification of their spectral properties has received limited attention. In this study, through a graph-theoretic formulation of river network topology, we investigate the eigenvalue spectra of its connectivity matrix (i.e., adjacency matrix). First, we explain the observed range of zero eigenvalues on the spectra using the notion of multiplicity (i.e., algebraic and geometric multiplicity) for both undirected and directed river networks. Next, we investigate the physical meaning of the multiplicity of zero eigenvalues on the dynamics of the river network. We show that multiplicity of zero eigenvalues is sufficient to determine the minimum set of driver nodes on the river network. The ratio of the number of driver nodes vs total number of nodes is a measurement of controllability of the river network, which is essential for a comprehensive understanding of the system's dynamics under external forcing. Using both synthetic and natural river networks, we show that with increasing heterogeneity, quantified via Tokunaga c-value, the number of zero eigenvalues increases indicating that basins in humid climate require more number of driver nodes to control their network dynamics. Finally, we show that driver nodes tend to avoid critical nodes identified via pairwise connectivity. Our results indicate that the multiplicity of zero eigenvalues in the eigenvalue spectrum can serve as a valuable tool for understanding and quantifying the physical and dynamical properties of river networks, such as controllability and heterogeneity. Furthermore, our findings establish a clear connection between controllability metrics and the vulnerability of river networks.