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.

翻译：我们研究了一个交互代理网络中的异步动力学，这些代理根据时变阈值规则更新其二元状态。具体而言，代理通过比较其交互网络中邻居当前状态的加权平均值与可能异构的时变阈值来异步修正其状态。这些阈值由外生信号决定，该信号代表一个外部影响场，模拟不同代理对某一状态相对于另一状态的偏好。我们证明了共识均衡（即所有代理具有相同状态的均衡）全局稳定性的充要条件，且该条件对外部场（恒定或时变）具有鲁棒性。我们的结果适用于一般加权有向交互网络，并基于某些网络协调博弈的超模性质，这些博弈的最佳反应动力学与线性阈值动力学一致。特别地，我们为这类博弈引入了鲁棒改进路径的新概念，并刻画了其存在性的条件。