The Spiral of Silence in Multi-agent Models for Opinion Formation

We generalize the classic multi-agent DeGroot framework for opinion dynamics by incorporating the Spiral of Silence theory from political science, which posits that individuals may withhold their opinions when they perceive them to be in the minority. As in the original DeGroot model, the social network is represented as a weighted directed graph encoding how agents influence one another. However, agents holding minority opinions become silent, meaning they do not express their views. We introduce two families of models. In Silence Opinion Memoryless (SOM^-) models, agents update their opinions by averaging those of their non-silent neighbors. In Silence Opinion Memory-based (SOM⁺) models, agents average the opinions of all neighbors, but for silent ones, only the most recently expressed opinion is used. We show that SOM^- models guarantee consensus on clique graphs but, unlike the classic DeGroot model, not on all strongly connected aperiodic graphs. For SOM⁺ models, even cliques may fail to reach consensus, illustrating that even minimal memory can significantly affect opinion dynamics. Finally, we validate our models through large-scale simulations on small-world networks with over two million agents. The results support the Spiral of Silence theory and reveal inherent limitations to consensus in more realistic settings.