# Distributed Averaging Problems over Directed Signed Networks

## TL;DR

This paper develops new distributed protocols for signed directed networks, establishing conditions for consensus and stability, and introduces fixed-time convergence methods using improved Laplacian functions, supported by Lyapunov analysis and examples.

## Contribution

It introduces novel Laplacian potential functions and protocols for signed directed networks, enabling consensus and fixed-time convergence analysis regardless of weight balance.

## Key findings

- Signed-average consensus occurs if and only if the network is structurally balanced.
- Fixed-time consensus can be achieved with nonlinear protocols.
- Lyapunov stability analysis links convergence to Laplacian properties.

## Abstract

This paper aims at addressing distributed averaging problems for signed networks in the presence of general directed topologies that are represented by signed digraphs. A new class of improved Laplacian potential functions is proposed by presenting two notions of any signed digraph: induced unsigned digraph and mirror (undirected) signed graph, based on which two distributed averaging protocols are designed using the nearest neighbor rules. It is shown that with any of the designed protocols, signed-average consensus (respectively, state stability) can be achieved if and only if the associated signed digraph of signed network is structurally balanced (respectively, unbalanced), regardless of whether weight balance is satisfied or not. Further, improved Laplacian potential functions can be exploited to solve fixed-time consensus problems of signed networks with directed topologies, in which a nonlinear distributed protocol is proposed to ensure the bipartite consensus or state stability within a fixed time. Additionally, the convergence analyses of directed signed networks can be implemented with the Lyapunov stability analysis method, which is realized by revealing the tight relationship between convergence behaviors of directed signed networks and properties of improved Laplacian potential functions. Illustrative examples are presented to demonstrate the validity of our theoretical results for directed signed networks.