Percolation behavior of partially interdependent networks with capacity and loads

Capacity-loaded networks with interdependent topologies accurately mirror various infrastructure networks. In this work, a partially interdependent network with capacity and loads model is proposed to portray the network structure in real systems. The theoretical framework based on percolation theory for predicting percolation thresholds in partially interdependent networks with capacity and loads is established using generating functions and self-consistent equations. The percolation transition of network is analyzed by initially removing $1-p$ fraction nodes and exploring the size of the giant component of the network after cascade failure. Random and scale-free networks are used for numerical and simulation experiments. We find that increasing the capacity parameter enhances the robustness of interdependent networks and alters the percolation characteristics within the network. The phase transition types in random networks exhibit notable variations across different average degrees, while those in scale-free networks are influenced by power-law exponents. Finally, the validity and accuracy of the proposed model is confirmed by a double-layer empirical network consisting of the World Cities Network and the U.S. Electricity Network.