Sentiment cycles in discrete-time homogeneous networks

•The paper approaches sentiment transition in a complex network.•Agents are classified in neutral, optimists and pessimists.•In continuous-time, the model delivers a stable steady-state outcome.•In discrete-time, stability holds under a homogeneous network of degree one.•Endogenous cycles emerge in discrete-time for a connectivity degree larger than one.

Consider a network connecting individual agents that are endowed with distinct sentiments or ‘views of the world’. Specifically, assume that each node in the network contains an agent that, at a given period tt, can be found in one of five states: sentiment neutrality, exuberant optimism, non-exuberant optimism, exuberant pessimism and non-exuberant pessimism. Local interaction rules, similar to those one encounters in rumor propagation models, make agents change their sentiment as they contact with others. Under a continuous-time framework, the proposed setting delivers a stable fixed-point equilibrium, meaning that the shares of agents in each sentiment category will converge to constant steady-state levels. The inspection of the same structure of analysis in discrete-time indicates that the stability outcome continues to hold when the connectivity degree is equal to 1. However, this result might change as one considers higher-order connectivity. In this last case, persistent endogenous waves of optimism and pessimism emerge under a reasonable parameterization of the model.