River networks in the framework of complex network theory

This study aims to conceptualize natural drainage networks as genuine complex networks. To overcome the limitation imposed by the regular connectivity structure of the traditional planted binary tree model-commonly employed to represent the planar configuration of fluvial systems-we introduce a line graph transformation. This mapping converts a planted binary tree into a plane tree, which can be regarded as a special class of loopless, directed graph characterized by inhomogeneous topology. From a geomorphological standpoint, our transformation naturally induces a hierarchical basin ordering that reflects the pioneering Gravelius stream order, offering insights into this scheme by projecting the hierarchy of tributary rivers onto a purely topological space. The application of complex network theory provides an explanatory perspective on river networks, yielding experimental findings that complement and extend established results in the geomorphological literature. In particular, our analysis of six catchments in southern Italy reveals previously unrecognized regularities, including a family of exact scaling laws and a surprisingly smooth width-type function that closely fits a Gamma distribution. These scaling laws emerge as power-law probability distributions for both the in-degree of nodes and the size of their in-components, which capture the scale invariance of tributary river lengths and drainage areas in tributary basins, respectively, thereby expanding the pantheon of known fluvial laws. Notably, these distributions represent sample distributions drawn from the complete populations of upstream lengths and drainage areas, with the sampling induced by our mapping through the selection of fluvial quantities corresponding to topologically well-defined elements of the plane tree. On the theoretical front, our approach is grounded in a model of directed, uncorrelated random graphs with arbitrary degree distributions. Among other things, by applying the generating function formalism to uncorrelated plane trees we derive an analytical expression for the distribution of drainage areas in tributary basins as a function of the distribution of tributary river lengths. By relying on average cluster properties, we predict the scaling exponent of the former distribution to be exactly 2 in the limit of large system sizes-a result that aligns well with both our empirical observations and previous findings in the complex network literature. Beyond the first-order approximation inherent in the model, our mapping also offers a coherent framework for interpreting the correlation structure consistently observed across all examined drainage networks.