Title: Optimal cost for strengthening or destroying a given network,
Abstract: 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.