In a network of agents forming opinions, each agent updates its belief by averaging the beliefs of its neighbors. This social averaging drives the network toward consensus — all agents eventually agree — but the consensus can be wrong. Without any connection to reality, the network converges to a weighted average of initial opinions, not to the truth. If every agent starts with a biased estimate, averaging transmits the bias throughout the network and crystallizes it into a shared incorrect belief.
Popescu, Syatriadi, and Vaidya (arXiv 2602.22627, February 2026) prove that a single informed agent — one that receives private feedback about the truth while also participating in social averaging — can pull an arbitrarily large network to the correct answer. One anchor suffices.
The mechanism requires two processes operating simultaneously. Every agent averages its neighbors' opinions, creating pressure toward consensus. The informed agent additionally receives private signals about the ground truth, creating pressure toward accuracy. These two pressures compound: the anchor moves toward truth; the anchor's immediate neighbors are pulled by the anchor; their neighbors are pulled by them; the correction propagates through the network like a wave of accuracy spreading from the single informed node.
The mathematical result is sharp. The network does not need multiple informed agents distributed throughout the graph. It does not need the informed agent to be highly connected or centrally located. The graph needs only to be “condensely anchored” — a structural property weaker than connectivity that ensures information from the anchor can reach every other agent through the averaging dynamics. On any graph satisfying this condition, regardless of the number of agents, the single anchor brings the entire network to the truth asymptotically.
The convergence mechanism operates through a two-step contraction that single-step analysis misses. At each step, the averaging process mixes opinions and the learning process corrects the anchor. Analyzed in a single step, neither process alone might show contraction toward truth — the averaging spreads error from uninformed agents, and the learning corrects only the anchor. But over two steps, the combination contracts: the anchor corrects in step one, and the averaging transmits the correction in step two. The mixed-operator-norm framework detects this two-step contraction that standard approaches overlook.
The result matters for understanding opinion formation in networks with unequal access to information. Most agents in most networks have no direct access to ground truth — they form opinions by talking to neighbors who formed opinions by talking to neighbors. The result says this indirect structure can still converge to truth, provided at least one agent somewhere in the network has access to reality and is connected to the rest. The network does not need many truth-tellers. It needs one, and a path for the truth to travel.