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

Abstract

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 BFACF Montecarlo algorithm developed for polymers, is used to probe the complex conformational substates structure. We characterize the average cost gains (and path-length variation) for increasing polymer density with respect to a Dijkstra routing and find a non-monotonic behavior as previously found in random networks. We observe the expected phase transition when switching from a convex to a concave cost function (e.g., $x^\gamma$, where $x$ represents the node overlap) and the emergence of ergodicity breaking, finally we show that the space of ground states for $\gamma<1$ is compatible with an ultrametric structure as seen in many complex systems such as some spin glasses.