The Price of Uncertainty for Social Consensus
Yunzhe Bai, Alec Sun
TL;DR
This work analyzes how uncertainty in observing neighbors affects the ability of a social network to reach consensus. By modeling uncertain observations as multiplicative perturbations $1+\varepsilon$ and using the price of uncertainty (PoU) to quantify worst-case social-cost inflation, the authors derive tight asymptotic bounds for consensus games: $\Theta(n^2 \varepsilon^2)$ under regimes $\varepsilon=\tilde{\Omega}(n^{-1/4})$. The main contributions are a strengthened lower bound via a refined initializer/gadget construction that amplifies the snowball effect of uncertainty, and a matching upper bound proven through a monovariant that bounds degree growth during uncertain best-response dynamics. The results resolve an open problem from Balcan et al. and provide sharp insights into how even small observation errors can drastically hinder consensus in large networks, with implications for designing robust social-influence processes and understanding biases in information aggregation.
Abstract
How hard is it to achieve consensus in a social network under uncertainty? In this paper we model this problem as a social graph of agents where each vertex is initially colored red or blue. The goal of the agents is to achieve consensus, which is when the colors of all agents align. Agents attempt to do this locally through steps in which an agent changes their color to the color of the majority of their neighbors. In real life, agents may not know exactly how many of their neighbors are red or blue, which introduces uncertainty into this process. Modeling uncertainty as perturbations of relative magnitude $1+\varepsilon$ to these color neighbor counts, we show that even small values of $\varepsilon$ greatly hinder the ability to achieve consensus in a social network. We prove theoretically tight upper and lower bounds on the price of uncertainty, a metric defined by Balcan et al. to quantify the effect of uncertainty in network games.