  Many systems such as critical infrastructure exhibit a modular structure with
many links within the modules and few links between them. One approach to
increase the robustness of these systems is to reinforce a fraction of the
nodes in each module, so that the reinforced nodes provide additional needed
sources for themselves as well as for their nearby neighborhood. Since
reinforcing a node can be an expensive task, the efficiency of the
decentralization process by reinforced nodes is vital. In our study we analyze
a new model which combines both above mentioned features of real complex
systems - modularity and reinforced nodes. Using tools from percolation theory,
we derived an analytical solution for any partition of reinforced nodes;
between nodes which have links that connect them to other modules
("inter-nodes") and nodes which have connections only within their modules
("intra-nodes"). Among our results, we find that near the critical percolation
point ($p\approx p_c$) the robustness is greatly affected by the distribution.
In particular, we find a partition of reinforced nodes which yields an optimal
robustness and we show that the optimal partition remains constant for high
average degrees.
