  Recent work on self-organized remodeling of vasculature in slime-mold, leaf
venation systems and vessel systems in vertebrates has put forward a plethora
of potential adaptation mechanisms. All these share the underlying hypothesis
of a flow-driven machinery, meant to alter rudimentary vessel networks in order
to optimize the system's dissipation, flow uniformity, or more, with different
versions of constraints. Nevertheless, the influence of environmental factors
on the long-term adaptation dynamics as well as the networks structure and
function have not been fully understood. Therefore, interwoven capillary
systems such as found in the liver, kidney and pancreas, present a novel
challenge and key opportunity regarding the field of coupled distribution
networks. We here present an advanced version of the discrete Hu--Cai model,
coupling two spatial networks in 3D. We show that spatial coupling of two
flow-adapting networks can control the onset of topological complexity in
concert with short-term flow fluctuations. We find that both
fluctuation-induced and spatial coupling induced topology transitions undergo
curve collapse obeying simple functional rescaling. Further, our approach
results in an alternative form of Murray's law, which incorporates local vessel
interactions and flow interactions. This geometric law allows for the
estimation of the model parameters in ideal Kirchhoff networks and respective
experimentally acquired network skeletons.
