Structural Robustness to Noise in Consensus Networks: Impact of Average Degrees and Average Distances

Abstract

We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We measure the structural robustness by using the smallest possible value of steady state population variance of states under the noisy consensus dynamics with edge weights from the unit interval. We derive tight upper and lower bounds on the structural robustness of networks based on the average distance between nodes and the average node degree. Using the proposed bounds, we characterize the networks with different types of robustness scaling under increasing size. Furthermore, we show that there is a fundamental trade-off between the structural robustness and the average degree of networks. We then show that, random k-regular graphs (the degree of each node is k) with n nodes typically have near optimal structural robustness among all the graphs with size n and average degree k. We also show that when k increases properly with n, random k-regular graphs maintain a structural robustness within a constant factor of the best possible value (corresponds to the complete graph) while also having the minimum average degree required for such robustness.

High Dimensional Robust Consensus Over Networks With Limited Capacity

“…It is possible to express H(G, w) in terms of the eigenvalues of L w (e.g., see [1,2]). In [7], it was shown that H(G, w) monotonically decreases in all edge weights and the smallest value under weights from the unit interval (0, 1] was defined as a structural measure of network robustness, i.e.,…”

“…, G k must all be tree graphs. For tree graphs, the unique minimizer of (2) is the star graph (e.g., see [7]). Accordingly, we will complete the proof by showing that when the capacity of each edge on a complete graph is 2α, it is possible to build all of G 1 , .…”

“…, G k must have n − 1 edges, namely each needs to be a tree. In light of [7,Thm 3.2], for any tree G ℓ , we have H * (G ℓ ) =δ(G ℓ )(n − 1)/4n, whereδ(G ℓ ) is the average distance of G ℓ . Accordingly, we are left to show that we can construct G 1 , .…”

“…Some graph-topological bounds on this robustness measure were presented in [5,6]. A structural robustness measure, which is the smallest possible value of steady state variance under edge weights from the unit interval, was presented along with some tight bounds in [7].…”

“…In this paper, we investigate the optimization of structural robustness to noise [7] in linear consensus networks under capacity-constrained communications. We model the capacity of each edge as an upper bound on the number of state vector components that can be instantaneously communicated.…”