A Mathematical Framework To Characterize Complexity Assembly In Fractal River Networks

As a multi-scale system featuring fractal hierarchical branching structure, the quantitative characterization of natural river networks is of fundamental significance for the assessment of the hydrological and ecological issues. However, as already evidenced, the fractal behavior cannot be uniquely inverted by fractal dimension, which induces a challenge in accurately describing the arbitrary scale-invariance properties in natural river networks. In this work, as per fractal topography theory, an open mathematical framework for the description of arbitrary fractal river networks is proposed by clarifying the assembly mechanisms of complexity types (i.e. the original and behavioral complexities) in a river network. On this basis, a general algorithm for the characterization of complexities is developed, and the effects of the original and behavioral complexities on the structure of a river network are systematically explored. The results indicate that the original complexity determines the tortuosity and spatial coverage of a river network, and the behavioral complexity dominates the river patterns, heterogeneity, and scale-invariance properties. Our investigation lays a foundation for assessing and predicting accurately the effect on environments, ecology and humans from river networks.