Naïve Learning with Uninformed Agents

The DeGroot model has emerged as a credible alternative to the standard Bayesian model for studying learning on networks, offering a natural way to model naïve learning in a complex setting. One unattractive aspect of this model is the assumption that the process starts with every node in the network having a signal. We study a natural extension of the DeGroot model that can deal with sparse initial signals. We show that an agent's social influence in this generalized DeGroot model is essentially proportional to the degree-weighted share of uninformed nodes who will hear about an event for the first time via this agent. This characterization result then allows us to relate network geometry to information aggregation. We show information aggregation preserves "wisdom" in the sense that initial signals are weighed approximately equally in a model of network formation that captures the sparsity, clustering, and small-world properties of real-world networks. We also identify an example of a network structure where essentially only the signal of a single agent is aggregated, which helps us pinpoint a condition on the network structure necessary for almost full aggregation. Simulating the modeled learning process on a set of real-world networks, we find that there is on average 22.4 percent information loss in these networks. We also explore how correlation in the location of seeds can exacerbate aggregation failure. Simulations with real-world network data show that with clustered seeding, information loss climbs to 34.4 percent.