Fig. 2: Graph set up to analyse the transition from tree networks to loopy networks.

a Elementary network to study spontaneous loop formation in optimum supply networks. The network consists of five nodes (green circles) where node n = 1 has an inflow of four, P₁ = 4, and all other nodes have an outflow of unity. These in and outputs determine the flows F_i, i ∈ {1, 2, 3, 4, 5} along with the links with capacities k_i. The optimum topology for this set-up is a tree network. If the in- and outputs are fluctuating, an additional edge (dotted arrow) may be beneficial to reduce the average dissipation. This edge introduces a new degree of freedom expressed as a cycle flow f. b For a larger network, we generalise this setup as follows: we start from a tree network and then consider the impact of a new edge at an arbitrary position (n, m) (dotted, red arrow). We then collect the edge sets L (shaded green) and R (shaded blue) along the shortest path from the source to the newly formed edge. This edge induces a cycle flow f. c A network formed from a triangular grid with a set of potential edges \({\mathcal{E}}\) coloured in grey which we will analyse throughout the manuscript. Realised edges (black) correspond to a global minimum of the dissipation for the fluctuating sink model where a single, fluctuating source (large circle) supplies the remaining network.