  The structure and design of optimal supply networks is an important topic in
complex networks research. A fundamental trait of natural and man-made networks
is the emergence of loops and the trade-off governing their formation: adding
redundant edges to supply networks is costly, yet beneficial for resilience.
Loops typically form when costs for new edges are small or inputs uncertain.
Here, we shed further light on the transition to loop formation. We demonstrate
that loops emerge discontinuously when decreasing the costs for new edges for
both an edge-damage model and a fluctuating sink model. Mathematically, new
loops are shown to form through a saddle-node bifurcation. Our analysis allows
to heuristically predict the location and cost where the first loop emerges.
Finally, we unveil an intimate relationship among betweenness measures and
optimal tree networks. Our results can be used to understand the evolution of
loop formation in real-world biological networks.
