Asynchronous Averaging on Dynamic Graphs with Selective Neighborhood Contraction

We study a discrete-time consensus model in which agents iteratively update their states through interactions on a dynamic social network. At each step, a single agent is selected asynchronously and averages the values of its current neighbors. A distinctive feature of our model is that an agent's neighborhood may contract following an update, while non-selected agents may add or remove neighbors independently. This creates a time-varying communication structure with endogenous contraction. We show that under mild assumptions--specifically, that the evolving graph is connected infinitely often--the system reaches consensus almost surely. Our results extend classical consensus theory on time-varying graphs and asynchronous updates by introducing selective neighborhood contraction, offering new insights into agreement dynamics in evolving social systems.