Resilient Consensus for Expressed and Private Opinions

This article proposes an opinion formation model featuring both a private and an expressed opinion for a given topic over dynamical networks. Each individual in the network has a private opinion, which is not known by others but evolves under local influence from the expressed opinions of its neighbors, and an expressed opinion, which varies under a peer pressure to conform to the local environment. We design the opinion sifting strategies which are purely distributed and provide resilience to a range of adversarial environment involving locally and globally bounded threats as well as malicious and Byzantine individuals. We establish the sufficient and necessary graph-theoretic criteria for normal individuals to attain opinion consensus in both directed-fixed and time-varying networks. Two classes of opinion clustering problems are introduced as an extension. By designing the resilient opinion separation algorithms, we develop necessary and sufficient criteria, which characterize the resilient opinion clustering in terms of the ratio of opinions as well as the difference of opinions. Numerical examples, including real-world jury deliberations, are presented to illustrate the effectiveness of the proposed approaches and test the correctness of our theoretical results.