Robust Coordination of Linear Threshold Dynamics on Directed Weighted Networks

We study 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 toward one state with respect to the other. We prove necessary and sufficient conditions for global stability of consensus equilibria, robustly with respect to the (constant or time-varying) external field. Our results apply to general weighted directed interaction networks and build on the supermodularity 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 the necessary and sufficient conditions for their existence.