A Manifold Minimization Principle for Physical Networks

A physical network is not only a combinatorial object, a graph of nodes and links, but also a geometric object that is best described as a smooth manifold, a "morphologically shaped" intrinsic d-dimensional space that is locally smooth everywhere. Inspired by string theory, we find that a graph can indeed be promoted to a smooth manifold when d≥2, by adding extra degrees of freedom that minimally describe the geometry of individual physical links (e.g., geodesic length and twist) and nodes (e.g., how links are sewed to each other). This manifold-based description further gives rise to a general manifold minimization principle for the formation of physical networks, minimizing not only the wiring length of all links (cf. the Steiner problem) but also other physical d-measures such as surface area or volume of the manifold. However, we show that in order to maintain the transportational functionality of each physical link, such d-measures cannot be minimized without constraint. Focusing on d=2, we introduce a systolic surface minimization scheme that keeps the shortest circumference (i.e., the systole) of each physical link fixed, finding that the solution predicts two novel structural transitions: a transition from bifurcation to trifurcation and a transition from sprouting-like bifurcation to branching-like bifurcation. Both transitions occur at finite length-systole ratios of the physical links and are forbidden by the Steiner solution, yet often observed in real biological systems, the transition thresholds agreeing with theoretical predictions. Our results reveal the inherent limitation of describing network geometry by shapeless 1D wires only, indicating a much richer geometry that pertains to the formation of manifold-described physical networks.