  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.
