A Time-Variant and Non-Linear Model of Opinion Formation in Social Networks

This paper develops a discrete-time, non-linear and time-variant model of opinion formation in a social network with global interactions to investigate the relationship between the final consensus belief and the set of agents' initial opinions. The model uses a novel and considerably intuitive updating rule, according to which the weight placed by an agent on another one's opinion in each period decreases continuously with the distance between their beliefs in the previous period. In this context, the first part of our analysis proves that agents' beliefs converge and reach a consensus over time (under a fairly general set of conditions). For the two-agent case, it is then shown that the consensus belief is the simple arithmetic mean of initial opinions. When there are three agents in the network, the combined use of computational and analytical methods reveals a relatively more complex polynomial relationship between long-run and initial beliefs. In particular, our results for the three agent-case imply that the deviation of the limiting belief from the corresponding average of initial beliefs can be expressed as a third-degree polynomial function incorporating the pairwise differences of agents' starting beliefs