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

Biological transport networks, such as those found in tree structures, can be characterized by a balance between viscous dissipation and metabolic cost. Traditionally, descriptors have either disregarded edge width or treated it as a hierarchical metric, but neither approach has effectively linked the network's structure to the dissipation-cost trade-off.

Researchers have now introduced the tree-to-cycle transition scale, a single numerical value derived from the radius of the thickest edge outside the maximum spanning tree. This scale can help determine the extent of redundancy a network tolerates. By applying this scale to networks adapting to spatially correlated load fluctuations, scientists found that the transition scale decreases linearly with dissipation and increases linearly with metabolic cost.

The scale exhibits the most significant scatter in the region where some networks fall outside the Pareto front. Adding a growth term in the equation removes the scatter but splits the correlation into two distinct branches. Essentially, the tree-to-cycle transition scale encapsulates a network's position on the Pareto front, which is influenced by the specific path the network follows during optimization.

This discovery provides a method for comparing various networks, like leaf venations in different species or growth conditions, based on the trade-off they achieve.