Physarum-based approach to distributed optimal transport on graphs

Biological transport networks achieve efficient transport while maintaining functionality under environmental changes and damage. Among them, the true slime mold Physarum polycephalum has attracted attention as it adaptively reorganizes its network structure based solely on local interactions. The Physarum model is known to be equivalent to optimal transport (OT) and has been applied in transport network studies. However, implementing it in artificial networks requires solving a graph-Laplacian linear system in a centralized manner, which makes local and distributed execution difficult in applications such as logistics systems and limits the exploitation of the environmental robustness inherent in slime mold behavior. Accordingly, we propose distributed Physarum-OT, a model that reflects the local and distributed nature of real slime mold. This method is formulated as a bilevel optimization problem wherein slime mold network development and the solution of the linear system are treated as the outer and inner problems, respectively. Furthermore, we propose a computational relaxation method with convergence guarantees. Numerical experiments on a logistics network show that the proposed method converges to OT in a locally distributed manner while adapting smoothly to cost changes in a slime-mold-like manner.