Optimal cost for strengthening or destroying a given network.

Strengthening or destroying a network is a very important issue in designing resilient networks or in planning attacks against networks, including planning strategies to immunize a network against diseases, viruses, etc. Here we develop a method for strengthening or destroying a random network with a minimum cost. We assume a correlation between the cost required to strengthen or destroy a node and the degree of the node. Accordingly, we define a cost function c(k), which is the cost of strengthening or destroying a node with degree k. Using the degrees k in a network and the cost function c(k), we develop a method for defining a list of priorities of degrees and for choosing the right group of degrees to be strengthened or destroyed that minimizes the total price of strengthening or destroying the entire network. We find that the list of priorities of degrees is universal and independent of the network's degree distribution, for all kinds of random networks. The list of priorities is the same for both strengthening a network and for destroying a network with minimum cost. However, in spite of this similarity, there is a difference between their p_{c}, the critical fraction of nodes that has to be functional to guarantee the existence of a giant component in the network.