Susceptibility optimization for wisdom of crowds: A social power reformulation and realization framework

Whether social influence makes a crowd wiser or more biased hinges on who yields to others and who holds firm. Such heterogeneous susceptibilities to social influence, together with the influence network, determine the weight each agent's initial estimate carries in the collective estimate, known as social power. This paper studies susceptibility optimization for maximizing the wisdom of crowds under the Friedkin-Johnsen model. Minimizing the squared collective bias over susceptibilities leads to a nonconvex problem. To overcome this nonconvexity, we develop a social power reformulation and realization framework, which decomposes the problem into two stages: first characterizing the social power that minimizes the collective bias, then determining the susceptibilities that induce such social power. Social power reformulation yields a quadratic program over the probability simplex. Under uniform-sign biases, the optimal social power concentrates on the least biased agents and immediately determines the optimal susceptibilities; under mixed-sign biases, its geometry enables a linear-time construction algorithm. For social power realization, we introduce the notions of susceptibility realizability and establish necessary and sufficient conditions for them, under which we derive all optimal susceptibilities in closed form using the Laplacian pseudoinverse and eigenvector centrality. These results yield a unified algorithm that solves the original problem globally, with worst-case complexity cubic in the number of agents when realizability holds. Finally, we develop a fully distributed implementation with linear communication complexity, illustrating how collective intelligence can emerge from local interactions.