Graph-Theoretic Robustness Analysis of Log Learning Learning Dynamics

We investigate the propagation of stubborn behavior in a network coordination game where players update their strategies using log-linear learning dynamics. A network is considered robust if the stubborn players cannot impact the stable behavior of the other players. We present a graph-theoretic framework for analyzing the robustness of various networks, establishing conditions wherein all network nodes switch to a stubborn behavior. Our framework leverages the notion of graph closed-knittedness, which measures the strength of external influence on a set of nodes. Using closed-knittedness and a closely related notion of graph plumpness, we derive necessary and sufficient conditions for network robustness to stubborn behavior. We validate our analytical results and bound through extensive Monte-Carlo simulations.