Tree-to-cycle transition scale characterizes trade-off between dissipation and construction cost in adaptive transport networks

Biological transport networks range from tree-like to highly reticulated architectures, which have been proposed to reflect different balances between viscous dissipation and metabolic cost. This balance cannot currently be inferred from structure: available descriptors either discard edge width entirely or preserve it only as a hierarchical decomposition that has not been mapped to the dissipation--cost trade-off. Here we introduce the tree-to-cycle transition scale, a single number given by the radius of the thickest edge outside the maximum spanning tree, which we interpret as the highest cost a network accepts for redundancy. We compute it for networks adapting to spatially correlated load fluctuations, whose correlation length sets their position on the Pareto front. The transition scale decreases linearly with dissipation and increases linearly with metabolic cost. Its scatter is largest in the regime where some networks end up off the front, and adding a growth term removes the scatter but splits the correlation into two branches. A single quantity read off the static network architecture thus characterizes where a network sits on the Pareto front, and is sensitive to the trajectory the network took through the optimization landscape. This suggests a route to comparing observed networks, for example leaf venations across species or growth conditions, by the trade-off they realize.