Emergence of algorithmically hard phases in transportation networks

We study a model of transportation networks with nonlinear elements which represent local shortage of resources. Frustration arises from competition among the nodes to become satisfied. When the initial resources are uniform, algorithmically hard regimes emerge when the average availability of resources increases. These regimes are characterized by discrete fractions of satisfied nodes, resembling the Devil's staircase. Behavior similar to those in the vertex cover or close packing problems are found. When initial resources are bimodally distributed, such algorithmically hard regimes also emerge when the fraction of rich nodes increases.