title: Optimization of the robustness of multimodal networks.
abstract: 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.
authors.affiliation: Kochi National College of Technology, Monobe-Otsu 200-1, Nankoku, Kochi 783-8508, Japan.
authors.forename: Toshihiro
authors.initials: T
authors.lastname: Tanizawa
authors.forename: Gerald
authors.initials: G
authors.lastname: Paul
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin
authors.forename: H Eugene
authors.initials: HE
authors.lastname: Stanley