Degree-Weighted DeGroot Learning

We examine belief formation in large stochastic networks under a degree-weighted DeGroot learning process. In many social settings, agents may place more trust in well-connected individuals or, conversely, discount their influence. Existing research on random networks primarily assumes that agents assign equal weights to their neighbors' opinions. We relax this assumption by allowing agents to weight neighbors based on their degree. Our main technical contribution is to derive the asymptotic properties of learning outcomes, particularly the consensus belief and convergence speed, under this degree-weighted framework. Using this result, we analyze how the weighting rule affects consensus beliefs, societal wisdom, and convergence speed. We find that a more popularity-favoring rule -- i.e., assigning greater weight to higher-degree neighbors -- harms wisdom but has a non-monotonic effect on convergence speed. Whether it accelerates or slows convergence depends on the diversity of views within high- and low-degree groups. This highlights a potential tradeoff between faster convergence and an unwise consensus in networks where agents favor highly connected neighbors.