Robust Coordination of Linear Threshold Dynamics on Directed Weighted Networks

TL;DR

This paper analyzes the stability of consensus in directed weighted networks where agents update binary states based on thresholds influenced by external signals, providing necessary and sufficient conditions for robust global stability.

Contribution

It introduces a comprehensive stability analysis for linear threshold dynamics on directed networks, including novel conditions and the concept of robust improvement paths.

Findings

Conditions for global stability of consensus are established.

The analysis applies to general directed weighted networks.

A new notion of robust improvement paths is introduced.

Abstract

We study asynchronous dynamics in a network of interacting agents updating their binary states according to a time-varying threshold rule. Specifically, agents revise their state asynchronously by comparing the weighted average of the current states of their neighbors in the interaction network with possibly heterogeneous time-varying threshold values. Such thresholds are determined by an exogenous signal representing an external influence field modeling the different agents' biases towards one state with respect to the other one. We prove necessary and sufficient conditions for global stability of consensus equilibria, i.e., equilibria where all agents have the same state, robustly with respect to the (constant or time-varying) external field. Our results apply to general weighted directed interaction networks and build on super-modularity properties of certain network coordination games whose best response dynamics coincide with the linear threshold dynamics. In particular, we introduce a novel notion of robust improvement paths for such games and characterize conditions for their existence.