# Reduction of Second-Order Network Systems with Structure Preservation

## TL;DR

This paper introduces a graph clustering-based framework for reducing second-order network systems while preserving their structural properties, ensuring the reduced model retains the original interconnection topology.

## Contribution

It presents a novel structure-preserving reduction method using graph clustering, generalizes controllability Gramian for second-order systems, and provides an efficient H2-norm computation approach.

## Key findings

- The method effectively preserves network structure in reduced models.
- The approach accurately approximates the original system with quantifiable error.
- Application to a small-world network demonstrates practical effectiveness.

## Abstract

This paper proposes a general framework for structure-preserving model reduction of a secondorder network system based on graph clustering. In this approach, vertex dynamics are captured by the transfer functions from inputs to individual states, and the dissimilarities of vertices are quantified by the H2-norms of the transfer function discrepancies. A greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters. Then, the reduced-order model is generated by the Petrov-Galerkin method, where the projection is formed by the characteristic matrix of the resulting network clustering. It is shown that the simplified system preserves an interconnection structure, i.e., it can be again interpreted as a second-order system evolving over a reduced graph. Furthermore, this paper generalizes the definition of network controllability Gramian to second-order network systems. Based on it, we develop an efficient method to compute H2-norms and derive the approximation error between the full-order and reduced-order models. Finally, the approach is illustrated by the example of a small-world network.