Dynamic centrality of headwater sources in river networks: a stochastic approach via ultrametric Laplacians

Abstract: River networks are hierarchical transport systems in which the timing and position of headwater confluences govern hydrologic response, solute transport, and ecological connectivity. Despite the recognized importance of headwater sources in structuring downstream processes, no mathematically grounded centrality index exists that captures their dynamic role in the transport hierarchy. We apply the dynamic centrality index $C_{\mathrm{CTMC}}$ [Morán Ledezma, arXiv:2603.20922], originally introduced in the context of phylogenetic trees, to the problem of headwater centrality in river networks via the dynamic tree representation of [Zaliapin et al., https://doi.org/10.1029/2009JF001281]. Through a topological analysis of the ultrametric structure induced by the dynamic tree, we show that high-centrality headwaters are the tributaries that most efficiently transmit water into the rest of the network, in the sense that their flows merge earliest and most broadly with surrounding sources as transport proceeds downstream. The index admits a fully explicit closed-form expression computable in $O(n)$ time from the tree structure alone, without simulation. Comparing $C_{\mathrm{CTMC}}$ rankings against the number of downstream junctions reached during transport, a direct measure of hydrological influence, on a dataset of 49 natural river basins across the United States, we find that top-ranked headwaters consistently reach a disproportionately large number of junctions across all transport times. This indicates that high-centrality headwaters are not merely early contributors but consistently influential throughout the entire transport process. These results suggest that ultrametric spectral analysis provides an interpretable and scalable framework for identifying hydrologically influential headwaters, with potential applications in ecological monitoring and watershed management.