[Paper Review] Gramian-Based Model Reduction of Directed Networks

Xiaodong Cheng, Jacquelien M.A. Scherpen|arXiv (Cornell University)|Jan 4, 2019

Model Reduction and Neural Networks39 references4 citations

TL;DR

This paper proposes a structure-preserving model reduction method for weakly connected directed network systems using pseudo Gramians to quantify vertex dissimilarity and guide graph clustering. By defining pseudo controllability and observability Gramians for semistable systems and leveraging them in a Petrov-Galerkin framework, the method generates reduced-order models that preserve the directed network topology and ensure bounded approximation error, with numerical results showing high accuracy even at low reduced orders.

ABSTRACT

This paper investigates a model reduction problem for linear directed network systems, in which the interconnections among the vertices are described by general weakly connected digraphs. First, the definitions of pseudo controllability and observability Gramians are proposed for semistable systems, and their solutions are characterized by Lyapunov-like equations. Then, we introduce a concept of vertex clusterability to guarantee the boundedness of the approximation error and use the newly proposed Gramians to facilitate the evaluation of the dissimilarity of each pair of vertices. An clustering algorithm is thereto provided to generate an appropriate graph clustering, whose characteristic matrix is employed as the projections in the Petrov-Galerkin reduction framework. The obtained reduced-order system preserves the weakly connected directed network structure, and the approximation error is computed by the pseudo Gramians. Finally, the efficiency of the proposed approach is illustrated by numerical examples.

Motivation & Objective

• To address the lack of structure-preserving model reduction methods for weakly connected directed networks, which are common in biochemical, ecological, and engineering systems.
• To develop a clustering strategy that ensures bounded approximation error by introducing the concept of vertex clusterability.
• To quantify vertex dissimilarity using pseudo Gramians derived from Lyapunov-like equations for semistable systems.
• To preserve the directed network structure in the reduced-order model through projections derived from graph clustering in the Petrov-Galerkin framework.
• To demonstrate the method's efficiency and accuracy on large-scale directed networks, including non-strongly connected examples where prior methods fail.

Proposed method

• Introduces pseudo controllability and observability Gramians for semistable systems, defined via solutions to Lyapunov-like equations.
• Defines vertex clusterability as a condition to guarantee bounded approximation error in model reduction.
• Uses the pseudo Gramians to compute pairwise dissimilarities between vertices, quantifying their dynamic behavior differences.
• Applies a graph clustering algorithm based on dissimilarity to group vertices into clusters, with the characteristic matrix of the clustering used as projection matrices.
• Applies the Petrov-Galerkin framework with these projections to derive a reduced-order model that preserves the Laplacian structure of the original directed network.
• Computes the $η_2$-norm approximation error using Theorem 8 to evaluate performance.

Experimental results

Research questions

• RQ1 How can model reduction be performed for weakly connected directed networks while preserving the network structure?
• RQ2 What criteria can ensure bounded approximation error when aggregating vertices in directed network systems?
• RQ3 How can vertex dissimilarity be quantified in directed networks with semistable dynamics?
• RQ4 Can pseudo Gramians of semistable systems be used effectively to guide graph clustering in model reduction?
• RQ5 How does the proposed method compare to input-only dissimilarity or random clustering in terms of approximation accuracy?

Key findings

• The proposed method achieves an $η_2$-norm approximation error of 0.0011 for a 100-vertex reduced model of a 735-vertex weakly connected directed network, indicating high accuracy.
• The approximation error decays rapidly as the reduced order increases, with both the proposed method and input-only dissimilarity-based method significantly outperforming random clustering.
• The majority of computational time (538.09 seconds) is spent solving the pseudo Gramian Lyapunov equations, while clustering and dissimilarity computation are much faster.
• The method is effective for non-strongly connected networks, where prior approaches relying on Frobenius eigenvectors fail, as demonstrated on a Harwell-Boeing matrix example.
• The reduced-order models preserve the directed network structure, as shown in Fig. 7, with the Laplacian matrix of the reduced system reflecting the original topology.
• The method remains computationally feasible for large-scale networks using standard solvers, though ADI or Krylov methods could be used for even larger systems.