Model Reduction and Clusterization of Large-Scale Bidirectional Networks

Abstract

This paper proposes two model reduction methods for large-scale bidirectional networks that fully utilize a network structure transformation implemented as positive tridiagonalization. First, we present a Krylov-based model reduction method that guarantees a specified error precision in terms of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal H}_{\infty}$</tex> </formula> -norm. Positive tridiagonalization allows us to derive an approximation error bound for the input-to-state model reduction without computationally expensive operations such as matrix factorization. Second, we propose a novel model reduction method that preserves network topology among clusters, i.e., node sets. In this approach, we introduce the notion of cluster uncontrollability based on positive tridiagonalization, and then derive its theoretical relation to the approximation error. This error analysis enables us to construct clusters that can be aggregated with a small approximation error. The efficiency of both methods is verified through numerical examples, including a large-scale complex network.