# Optimizing the Coherence of Composite Networks

## TL;DR

This paper investigates how to connect multiple disjoint networks to optimize overall system coherence, providing analytical formulas, bounds, and algorithms for different consensus dynamics scenarios.

## Contribution

It introduces new analytical expressions and algorithms for optimizing the coherence of composite networks under various noisy consensus dynamics.

## Key findings

- Derived formulas for coherence based on individual networks and interconnections.
- Identified optimal interconnection topologies for coherence maximization.
- Developed a non-combinatorial algorithm for near-optimal network connection.

## Abstract

We consider how to connect a set of disjoint networks to optimize the performance of the resulting composite network. We quantify this performance by the coherence of the composite network, which is defined by an $H_2$ norm of the system. Two dynamics are considered: noisy consensus dynamics with and without stubborn agents. For noisy consensus dynamics without stubborn agents, we derive analytical expressions for the coherence of composite networks in terms of the coherence of the individual networks and the structure of their interconnections. We also identify optimal interconnection topologies and give bounds on coherence for general composite graphs. For noisy consensus dynamics with stubborn agents, we develop a non-combinatorial algorithm that identifies connecting edges such that the composite network coherence closely approximates the performance of the optimal composite graph.