Many technological networks can experience random and/or systematic failures in their components. More destructive situations can happen if the components have limited capacity, where the failure in one of them might lead to a cascade of failures in other components, and consequently break down the structure of the network. In this paper, the tolerance of cascaded failures was investigated in weighted networks. Three weighting strategies were considered including the betweenness centrality of the edges, the product of the degrees of the end nodes, and the product of their betweenness centralities. Then, the effect of the cascaded attack was investigated by considering the local weighted flow redistribution rule. The capacity of the edges was considered to be proportional to their initial weight distribution. The size of the survived part of the attacked network was determined in model networks as well as in a number of real-world networks including the power grid, the internet in the level of autonomous system, the railway network of Europe, and the United States airports network. We found that the networks in which the weight of each edge is the multiplication of the betweenness centrality of the end nodes had the best robustness against cascaded failures. In other words, the case where the load of the links is considered to be the product of the betweenness centrality of the end nodes is favored for the robustness of the network against cascaded failures.
The complexity of the optimal phase control problem in wireless MIMO systems with scalar feedback quantization and equal-gain transmission is studied. The problem is shown to be NP-hard when the number of receive antennas grows linearly with the number of transmit antennas. For the case where the number of receive antennas is constant, the problem can be solved in polynomial time. An optimal algorithm is explicitly constructed. For practical purposes, a low-complexity algorithm based on local search is presented. Simulation results show that its performance is nearly optimal.
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