The ControlRank index is a spectral measure designed to quantify node importance in both static and dynamic networks under control settings [2]. ControlRank is based on linear control theory and evaluates the influence of a node by analyzing the minimum eigenvalues of matrices derived from the network's Laplacian. A larger minimum eigenvalue indicates that the node contributes more to the network’s overall stability and accelerates convergence dynamics.The ControlRank index of node \(i\) is defined as\[c_{ControlRank}(i) = λ_i(L^s),\]where \(λ_i(L^s)\) is the minimum eigenvalue of the principal minor of \((L + L^T)/2\) obtained by removing the \(i\)-th row and column from the Laplacian matrix \(L\).

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Zhou, J., Yu, X., & Lu, J. A. (2018). Node importance in controlled complex networks. IEEE Transactions on Circuits and Systems II: Express Briefs, 66(3), 437-441. doi: 10.1109/TCSII.2018.2845940.