Biharmonic Distance-Based Performance Metric for Second-Order Noisy Consensus Networks

Abstract

We study second-order consensus dynamics with random additive disturbances. To quantify the robustness of these networks, we investigate three different performance measures: the steady-state variance of pairwise differences between vertex states, the steady-state variance of the deviation of each vertex state from the average, and the total steady-state variance of the system. We show that these performance measures are closely related to the concept of biharmonic distance; the square of the biharmonic distance plays a similar role in the system performance as resistance distance plays in the performance of first-order noisy consensus dynamics. We then define the new concepts of biharmonic Kirchhoff index and vertex centrality based on the biharmonic distance. We further derive analytical results for the performance measures and concepts for complete graphs, star graphs, cycles, and paths, and we use this analysis to compare the asymptotic behavior of the steady-state variance in first-and second-order systems. Finally, we propose a theoretically guaranteed approximation algorithm to estimate the total steady-state variance, which has a complexity of nearly linear time with respect to the number of edges. Extensive experiments results validate both efficiency and accuracy of our algorithm.