Optimal Location of Sources in Transportation Networks

We consider the problem of optimizing the locations of source nodes in transportation networks. A reduction of the fraction of surplus nodes induces a glassy transition. In contrast to most constraint satisfaction problems involving discrete variables, our problem involves continuous variables which lead to cavity fields in the form of functions. The one-step replica symmetry breaking (1RSB) solution involves solving a stable distribution of functionals, which is in general infeasible. In this paper, we obtain small closed sets of functional cavity fields and demonstrate how functional recursions are converted to simple recursions of probabilities, which make the 1RSB solution feasible. The physical results in the replica symmetric (RS) and the 1RSB frameworks are thus derived and the stability of the RS and 1RSB solutions are examined.