Non-Consensus Opinion Models with Byzantine Nodes

Abstract

Opinion dynamics models study how the interaction among people influences the opinion formation process. In most opinion dynamics models, only one opinion could exist in the steady state, which is different from the real-life opinion formation process. In 2009, Shao \emph{et al.} introduced a non-consensus opinion (NCO) model, which allows different opinions to coexist in the steady state. This thesis extends the NCO model by introducing a special type of nodes, Byzantine nodes, to play the role of dishonest people. The Byzantine NCO model is more in line with the real-world opinion formation process because it considers that people who express opinions are not always honest. I build an NCO model simulation algorithm and use this algorithm to perform simulations on three different network models: small-scall graphs, the Erdős–Rényi random graph and the scale-free network. In Byzantine node selection, three different strategies are proposed, according to the degree of the selected nodes. I find a new steady state for the NCO model: the cyclic steady state. The cyclic behaviour of the NCO and Byzantine NCO model is discussed, and some networks with a long cycle period are given. I also introduced a general method to generate networks with extremely long cycle periods. The other properties of the Byzantine NCO model, such as the probability of cyclic behavior, the final opinion distribution and the convergence time are researched. By performing simulations on the network models, I find that the introduction of Byzantine nodes could help the system to reach a steady state with a more balanced opinion ratio. The introduction of Byzantine nodes could decrease the critical threshold of the NCO model and promote the coexistence steady state. A mechanism in which Byzantine nodes influences the convergence time by influencing the steady state is suggested.