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, P(k)i=1 m a-(i-1) δ(k- ki) with ki b-(i-1) and Dirac's delta function δ(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 N1, that (i) the robustness against random and targeted node removal for this multimodal network is controlled by a single combination of variables, N1(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 ln N, and (iii) the values of exponent λopt 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.