Directed Weighted (?)-index And Its Application

Complex network theory is widely used in system optimization,prevention and control of infectious diseases,network dynamics,and other fields.As an important branch of the complex network,the H-type index is of great significance to network optimization,disaster avoidance,and control.In recent years,various H-type indices have been proposed one after another,which greatly enriches the theory of identifying the network node importance.However,few H-indices are considered comprehensively.Therefore,this paper proposes a new directed weighted h-index(abbreviated as dw-(?) index)to measure the importance of nodes in the directed weighted network and identify the vulnerability of lines in the power grid.The degree and strength of nodes,the weight and direction of edges,and the importance of neighbor nodes are considered in the dw-(?) index.The product of directed weight and directed strength of nodes is used to calculate the dw-(?) index.According to the iterative property of the operator H,the dw-(?) index of synchronous iteration and asynchronous iteration is introduced.Dw-(?) index can effectively measure the importance of nodes in directed weighted networks.The dw-(?) index can be iterated by the operator H to obtain the dw-(?) index sequence.It is proved theoretically that dw-(?) index sequences converge to the same value after finite step iteration in the process of synchronous update and asynchronous update.The analysis based on the citation network shows that the convergence speed of the dw-(?) index iterative sequence is very fast.In small networks or local networks,the first-order dw-(?) index can reach the limit of dw-(?) index approximately.It uses the information of two-step local networks centered on the target nodes and only needs to apply the dw-(?) index algorithm once.In large-scale networks and dynamic networks,the asynchronous update can be used to flexibly select the number of nodes in each iteration.Compared with the synchronous update,it can greatly shorten the convergence time of the dw-(?) index and save the cost of computing time.To effectively predict cascading failures and prevent large-scale blackouts,generally speaking,we don’t need to pay consideration to all the transmission lines in the power grid,we only need to pay consideration to some vulnerable transmission lines.For this reason,we transform the power grid into the well-known directed weighted network(i.e.correlation network)and transform the vulnerability assessment of transmission lines into the importance assessment of nodes.In the existing research,we improve the first-order and second-order correlation network,establish the third-order and multi-order correlation network,to further grasp the dynamic relationship between the branches of the power grid.In our third-order and multi-order correlation networks,the dw-(?) index algorithm is used to identify the important nodes in the correlation network,that is,the vulnerable lines in the power grid.The analysis results of the IEEE-24 bus system show that the accuracy of the dw-(?) index is more than 13.33%higher than that of existing methods.Therefore,the dw-(?) index is not only suitable for citation networks,but also has great application prospects in the power grid and other general directed weighted networks.