# Efficient Computation of H2 Performance on Series-Parallel Networks

## TL;DR

This paper presents an efficient method for computing the $ ext{H}_2$ performance measure on series-parallel networks by leveraging network decomposition, and also addresses adaptive re-weighting to optimize this measure.

## Contribution

It introduces a novel approach for distributed $ ext{H}_2$ norm computation on series-parallel networks using decomposition and composition rules, and extends to adaptive re-weighting for optimization.

## Key findings

- Efficient $ ext{H}_2$ norm computation via network decomposition.
- Distributed algorithms for re-weighting to optimize $ ext{H}_2$ performance.
- Complexity of adaptive re-weighting comparable to initial computation.

## Abstract

Series-parallel networks are a class of graphs on which many NP-hard problems have tractable solutions. In this paper, we examine performance measures on leader-follower consensus on series-parallel networks. We show that a distributed computation of the $\mathcal{H}_2$ norm can be done efficiently on this system by exploiting a decomposition of the network into atomic elements and composition rules. Lastly, we examine the problem of adaptively re-weighting the network to optimize the $\mathcal{H}_2$ norm, and show that it can be done with similar complexity.