Symmetry breaking in minimum dissipation networks

Abstract

Both natural and engineered supply networks exhibit universal structural patterns, such as the formation of loops, yet the principles governing optimal structures remain unclear. These patterns can be interpreted as solutions of optimization models, assuming that biological networks evolve toward optimal states and engineered systems are designed accordingly. We study a canonical model of transport networks that minimizes dissipation under a global resource constraint and admits analytical treatment. Symmetry breaking in optimal networks occurs in two distinct forms: weak symmetry breaking, which preserves symmetry in the topology but breaks it in the edge weights, and strong symmetry breaking, which eliminates it entirely. Varying the resource scaling exponent induces discontinuous transitions between these states and the fully symmetric phase, either through bifurcations of local minima or through exchanges of stability between competing optima. Moreover, fluctuations play a nontrivial role: as noise increases, the system can undergo a reentrant transition from strongly symmetry-broken to symmetric and back, implying the existence of an optimal fluctuation level that stabilizes symmetric network structures. These mechanisms persist beyond the minimal model. In a renewable energy network, fluctuations similarly influence optimal topologies. These results establish symmetry breaking as a generic organizing principle of optimal transport networks.