Anchoring and Mixed-Norm Contractions in Averaging-Learning Dynamics

Abstract

A single informed agent can draw an arbitrarily large network to the ground truth. This is the sharpest consequence of the "Averaging plus Learning" framework studied here, where agents update opinions by socially averaging neighbours while some receive private feedback at heterogeneous rates. The key is a graph-theoretic property we call condensely anchored, which implies convergence to the correct consensus on fixed networks. In the original framework of Popescu and Vaidya (2023), every agent was required to learn. Removing that requirement changes the problem fundamentally: the underlying graph must now carry the signal from a handful of anchors to everyone else. When learning rates decay to zero, a persistence condition on the rates alone suffices, with no uniform connectivity or aperiodicity assumed. The hardest case is intermittent connectivity, where no single time step contracts in any standard norm. A mixed-operator-norm framework is developed that extracts two-step contraction from the interplay between aggregate learning mass and entrywise diffusion of influence, a mechanism new to consensus literature. Finally, we demonstrate the framework's robustness: vanishing noise preserves convergence to the ground truth, whereas persistent noise drives the system to a limiting law.