Optimization of robustness based on reinforced nodes in a modular network

Many networks such as critical infrastructures exhibit a modular structure. One approach to increase the robustness of these systems is to reinforce a fraction of the nodes so that the reinforced nodes provide additional needed sources for themselves as well as for their nearby neighborhood. Since reinforcing a node can be expensive, the efficiency of the decentralization process by reinforced nodes is vital. Here we develop a model which combines both modularity and reinforced nodes and study the robustness of the system. Using tools from percolation theory, we derive an analytical solution for the robustness resulting from any partition of reinforced nodes; between nodes that have links that connect between modules and nodes which have links only within modules. We find that near the critical percolation threshold the robustness is greatly affected by the partition. In particular, we find a partition of reinforced nodes that yields optimal robustness and we show that the optimal partition remains constant for high average degrees.