Up to this day, resistance distance has many important applied areas, such as phase transition, silicate network, electrical network, and complex networks. extended the resistance distance to the weighted graph. Combining the normalized Laplacian matrix and the transition probability matrix, Chen and Zhang proposed a new degree-Kirchhoff index.
Xiao and Gutman uncovered some properties of a resistance matrix. Undoubtedly, further analysis of mixed networks has become an indispensable topic in the research of network science and interdisciplinary.ġ993, Klein and Randić first proposed resistance distance. Many efficient algorithms have been proposed to resolve the function and topological structure of mixed networks, such as decentralized optimization, node anomaly detection algorithm, embedding algorithm, and magnetic Laplacian. Mixed networks, just like the underlying one, emerge in the fields of multiagent systems, biology, and society. Many real-world networks are mixed, such as citation networks, web page link networks, e-mail networks, and information diffusion networks. Network science has been an emerging area in big data, cyberspace security, and artificial intelligence. Finally, we compare the results with the algebraic connectivity to verify the superiority of the proposed strategy. Meanwhile, we also propose a modified algorithm, the directed traversal algorithm, to select the edges whose removal will maximize the Hermitian Kirchhoff index in the general mixed graph. In this paper, we propose the positive walk algorithm to construct the Hermitian matrix for the mixed graph and then introduce the Hermitian resistance matrix and the Hermitian Kirchhoff index which are based on the eigenvalues and eigenvectors of the Hermitian Laplacian matrix. There are various measures of network criticality which have been investigated for underlying networks, while little is known about the corresponding metrics for mixed networks. The superiority of resistance distance and Kirchhoff index is that it can reflect the global properties of the graph fairly, and they are widely used in assessment of graph connectivity and robustness. The information-theoretic distance measure, namely, resistance distance, is a vital parameter for ranking influential nodes or community detection. Large-scale social graph data poses significant challenges for social analytic tools to monitor and analyze social networks.
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