Weighted Directed $\hbar$-Index and Its Application

This article proposes a new weighted directed $\hbar$-index to identify vulnerable lines in power grids and important nodes in directed weighted networks. This index iteratively uses more network information to update the importance of nodes in the network. Under some certain conditions, the weighted directed $\hbar$-index is more robust than the $h$-degree and the $w$-lobby index. The problem of identifying vulnerable lines in the power grid is transformed into the identification of important nodes in the directed weighted network. It is found that using proposed weighted directed $\hbar$-index (i.e. WDH3) on the top 20 transmission lines of the $IEEE$-118 bus system improves the recognition accuracy by at least 20% compared to some other existing methods. In the empirical network, the given weighted directed $\hbar$-index is also shown as an excellent technique for identifying key nodes in the network.