Reduced Order Modeling of Linear Consensus Networks using Weight Assignments

Abstract

This paper studies a model reduction method for linear consensus networks consisting of diffusively coupled single-integrators. For a given graph clustering of an original complex network, we construct a simplified network consisting of fewer nodes, where the edge weights are to be determined. An optimal weight assignment procedure is proposed to select suitable edge weights of the reduced network, aiming for the minimum H2 approximation error between the original network and the reduced-order network model. The effectiveness of the proposed method is illustrated by means of an example.

Model Reduction Methods for Complex Network Systems

“…Although an almost equitable partition as a particular clustering offers us analytical expression for the reduction error, it does not necessary lead to a small error. In fact, the methods in (65,66) provide significantly lower errors via alternative choices of clustering for some examples. Moreover, how to find all almost equitable partitions for a large-scale graph is generally a rather difficult and computationally expensive problem (25).…”

11 Reduced-order modeling of large-scale network systems

“…Moreover, F ∈ ℝ n×p in (11.14) is the binary matrix such that [F] ij = 1 if vertex i is the j-th leader, and [F] ij = 0 otherwise. Assume that the output of (11.14) is given as 25) where R is the incidence matrix of and W is the edge weight matrix defined in (11.10). Then, the output of the reduced network model (11.19) is obtained aŝ y =Ĥx = W an explicit ℋ 2 -error can be derived, which is characterized by the cardinalities of the clusters containing leaders [58,46].…”

“…Generally, all the existing clustering-based reduction methods fall into the framework of Petrov-Galerkin projections. In [25,24], an ℋ 2 -optimal approach is presented, which does not aim to find a suitable graph clustering. Instead, this approach focuses on how to construct a "good" reduced-order model for a given clustering.…”

“…Ifŵ is initialized as the outcome of clustering-based projection methods, the approximation accuracy obtained by the edge weighting approach will be better than the ones obtained by clustering-based projection after iteration. Furthermore, to solve the optimization problem in (11.49), we can also use a cross-iteration algorithm presented in [25]. Solve (11.51) to obtain the optimal solutionŵ * .…”

“…Subsequent research in [12,22,17,19] shows that the dissimilarity-based clustering method can also be extended to second-order networks, controlled power networks, and directed networks. In [25,24], a framework is presented on how to build a reduced-order model from a given clustering. The edge weights in the reduced network are parameterized so that an optimization problem is formulated to minimize the reduction error.…”