Biharmonic Distance and Performance of Second-Order Consensus Networks with Stochastic Disturbances

Abstract

We study second order consensus dynamics with random additive disturbances. 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 biharmonic distance; the square of the biharmonic distance plays similar role in the system performance as resistance distances plays in the performance of first-order noisy consensus dynamics. We further define the new concepts of biharmonic Kirchhoff index and vertex centrality based on the biharmonic distance. Finally, we 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-variance in first-and second-order systems.

Coherence Scaling of Noisy Second-Order Scale-Free Consensus Networks

“…The sum of biharmonic distances over all the N (N −1)/2 pairs of vertices in graph G is called its biharmonic index [26], denoted by B(G). Similarly to R(G), B(G) can be represented in terms of the N − 1 non-zero eigenvalues of L:…”

“…Similarly to H FO , H SO is completely determined by the nonzero eigenvalues of the Laplacian matrix [24]. Specifically, H SO is determined by the biharmonic index of the network [26]:…”

“…Relatively to the first-order case, related works about second-order network coherence H SO are much less, with the exception of few particular graphs, such as tori [24], fractal trees [25], Koch networks [31], hierarchical graphs [19] and Sierpiński graphs [19]. Very recently, Yi, Zhang, and Patterson [26] established a connection between the biharmonic distance of a graph and its second-order network coherence. They provided exact solutions to second-order network coherence of complete graphs, star graphs, cycles, and paths.…”

“…In addition to convergence rate and time delay, many other interesting quantities about consensus dynamics are also governed by the eigenvalues of the graph Laplacian matrix L. For example, for first-and second-order noisy networks without leaders, their network coherence defined in terms of the system's H 2 -norm (i.e., the average of deviations of agent states from the current average value) is determined by the sum of the reciprocal of the square of each nonzero eigenvalue of L [22][23][24][25][26]. For the first-order consensus problem, the network coherence has been studied for graphs with different structures, including paths [27], stars [27], cycles [27], Vicsek fractal trees and T-fractal trees [25,28], tori and lattices [24], Farey graphs [29,30], Koch networks [31], hierarchical graphs [19], Sierpinśki graphs [19], and some real-world networks [32].…”

Multi-Feature Fusion Based on Multi-View Feature and 3D Shape Feature for Non-Rigid 3D Model Retrieval

“…To overcome the above shortcomings of the geodesic distance, we use the biharmonic distance to measure the distance between the 3D surface points. The biharmonic distance is an effective surface distance measurement based on the biharmonic differential operator, which applies different weights to the eigenvalues of the Laplace-Beltrami operator [30], [31]. It provides a nice trade-off between nearly geodesic distances for small distances and global shape-awareness for large distances.…”