Diffusion-localization transition caused by nonlinear transport on complex networks

We analyzed nonlinear transport as defined for directed complex networks, where the flux from one node to a neighboring node is given preferentially according to the scalar quantities at the neighbor nodes. This is known as the generalized gravity interaction. In our research, we discovered a novel phase transition type. In the diffusion phase, the scalar quantity is scattered over the whole system, whereas in the localization phase, the flow tends to form localized confluence patterns owing to nonlinearity, resulting in the appearance of special nodes that irreversibly attract huge amounts of flow. We analytically considered the transition for selected network configurations, demonstrating that the transition point depends on the network topology. We also demonstrated that the diffusion phase of this transport model fits well with data from business firms, implying that the whole network structure can be used to model money flow in the real world.