Stochastic stability of continuous time consensus protocols

TL;DR

This paper presents a comprehensive framework for analyzing the convergence and stochastic stability of continuous time consensus protocols in complex networks, emphasizing spectral and geometric graph properties.

Contribution

It introduces a unified approach applicable to directed, time-varying, and stochastic networks, highlighting the role of spectral graph theory in stability and design.

Findings

Spectral properties like algebraic connectivity influence stability.

Expander graphs maintain high performance as network size grows.

Random topologies can be advantageous for consensus protocol design.

Abstract

A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and those whose topology and strength of connections may vary in time. The graph theoretic interpretation of the analytical results is emphasized. We show how the spectral properties, such as algebraic connectivity and total effective resistance, as well as the geometric properties, such the dimension and the structure of the cycle subspace of the underlying graph, shape stability of the corresponding CPs. In addition, we explore certain implications of the spectral graph theory to CP design. In particular, we point out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly. Similarly, we highlight the benefits of using random versus regular network topologies for CP design. We illustrate these observations with numerical examples and refer to the relevant graph-theoretic results. Keywords: consensus protocol, dynamical network, synchronization, robustness to noise, algebraic connectivity, effective resistance, expander, random graph