Optimal Location of Sources in Transportation Networks

TL;DR

This paper addresses the complex problem of optimally placing source nodes in transportation networks by developing a feasible 1RSB solution through functional recursion simplification, revealing phase transitions and solution stability.

Contribution

It introduces a novel method to convert functional cavity fields into simple probability recursions, enabling practical 1RSB analysis of continuous-variable network optimization.

Findings

Identification of a glassy transition related to surplus node fraction reduction

Development of a simplified recursive approach for 1RSB solutions

Analysis of stability for RS and 1RSB solutions in the model

Abstract

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.