Automorphism Groups and Multiscale Symmetry Breaking in a Rooted Watershed Tree: An Algebraic–Geometric Study of the Funa Drainage Network

Introduction: Drainage networks can be represented as rooted planar trees, making it possible to study their branching structures through algebraic and geometric concepts of symmetry. This study models the Funa drainage network in Kinshasa as a rooted spatial tree and investigates its exact automorphisms, approximate branch symmetries, and multiscale symmetry breaking.

Methods: The drainage network was represented as a directed rooted tree, with the outlet as the root, river reaches as edges, and sources and confluences as vertices. Exact symmetries were identified through rooted-subtree isomorphisms and computation of the automorphism group. Approximate symmetry at each binary confluence was quantified using a composite index, combining topological similarity, branch-length similarity, and angular similarity. Symmetry was then analysed across hierarchical scales defined by the number of contributing sources and Strahler stream order.

Results: The validated tree contains 66 vertices, 65 edges, 33 sources, and 32 binary confluences. Eight confluences, representing 25 percent of all confluences, exhibit exact topological symmetry. Seven correspond to exchanges between terminal-source branches, while one represents a higher-order branched symmetry. The rooted automorphism group contains 256 elements. Twenty-one exact rooted-subtree isomorphism classes were identified, including seven recurrent classes. No symmetry remained exact after strict metric and angular constraints were imposed. The mean approximate-symmetry score was 0.528, while the source-weighted global score was 0.405, producing a symmetry-breaking index of 0.595. Symmetry decreased significantly with increasing hierarchical scale, with a Spearman correlation coefficient of minus 0.737 and a probability value of 0.0000015.

Conclusions: The Funa network exhibits recurrent local automorphisms within a globally asymmetric hierarchy. The proposed framework provides a reproducible algebraic and geometric method for analysing symmetry, subtree equivalence, and scale-dependent symmetry breaking in hydrographic networks.