Optimization of the robustness of multimodal networks.

We investigate the robustness against both random and targeted node removal of networks in which P(k), the distribution of nodes with degree k, is a multimodal distribution, [formula--see text] with k(i) proportional to b -(i-1) and Dirac's delta function delta (x). We refer to this type of network as a scale-free multimodal network. For m=2, the network is a bimodal network; in the limit m approaches infinity, the network models a scale-free network. We calculate and optimize the robustness for given values of the number of modes m, the total number of nodes N, and the average degree <k>, using analytical formulas for the random and targeted node removal thresholds for network collapse. We find, when N>>1, that (i) the robustness against random and targeted node removal for this multimodal network is controlled by a single combination of variables, N(1/m-1), (ii) the robustness of the multimodal network against targeted node removal decreases rapidly when the number of modes becomes larger than a critical value that is of the order of 1n N, and (iii) the values of exponent lambda(opt) that characterizes the scale-free degree distribution of the multimodal network that maximize the robustness against both random and targeted node removal fall between 2.5 and 3.