Ultrametricity of optimal transport substates for multiple interacting paths over a square lattice network

We model a set of point-to-point transports on a network as a system of polydisperse interacting self-avoiding walks (SAWs) over a finite square lattice. The ends of each SAW may be located both at random, uniformly distributed, positions or with one end fixed at a lattice corner. The total energy of the system is computed as the sum over all SAWs, which may represent either the time needed to complete the transport over the network, or the resources needed to build the networking infrastructure. We focus especially on the second aspect by assigning a concave cost function to each site to encourage path overlap. A simulated annealing optimization, based on a modified Berg-Foerster-Aragao de Carvalho-Caracciolo-Froehlich (BFACF) algorithm developed for polymers, is used to probe the complex conformational substate structure at zero temperature. We characterize the average cost gains (and path-length variations) for increasing polymer density with respect to a Dijkstra routing and find a nonmonotonic behavior as recently found for random networks. We observe the emergence of ergodicity breaking and of nontrivial overlap distributions among replicas when switching from a convex to a concave cost function (e.g., x^{γ}, where x represents the node overlap). Finally, we show that the space of ground states for γ<1 is compatible with an ultrametric structure, as seen in many complex systems such as some spin glasses.