How robust is the wisdom of the crowds?

We introduce the study of adversarial effects on wisdom of the crowd phenomena. In particular, we examine the ability of an adversary to influence a social network so that the majority of nodes are convinced by a falsehood, using its power to influence a certain fraction, μ < 0.5 of N experts. Can a bad restaurant make a majority of the overall network believe in the quality of that restaurant by misleading a certain share of food critics into believing its food is good, and use the influence of those experts to make a majority of the overall network to believe in the quality of that restaurant? We are interested in providing an agent, who does not necessarily know the graph structure nor who the experts are, to determine the true value of a binary property using a simple majority. We prove bounds on the social graph's maximal degree, which ensure that with a high probability the adversary will fail (and the majority vote will coincide with the true value) when it can choose who the experts are, while each expert communicates the true value with probability p > 0.5. When we examine expander graphs as well as random graphs we prove such bounds even for stronger adversaries, who are able to pick and choose not only who the experts are, but also which ones of them would communicate the wrong values, as long as their proportion is 1 - p. Furthermore, we study different propagation models and their effects on the feasibility of obtaining the true value for different adversary types.