Phase Transition of a Non-Linear Opinion Dynamics with Noisy Interactions

Description

In several real Multi-Agent Systems (MAS), it has been observed that only weaker forms of metastable consensus are achieved, in which a large majority of agents agree on some opinion while other opinions continue to be supported by a (small) minority of agents. In this work, we take a step towards the investigation of metastable consensus for complex (non-linear) opinion dynamics by considering the popular Undecided-State dynamics in the binary setting, which is known to reach consensus exponentially faster than the Voter dynamics. We propose a simple form of uniform noise in which each message can change to another one with probability $p$ and we prove that the persistence of a metastable consensus undergoes a phase transition for $p= \frac{1}{6}$. In detail, below this threshold, we prove the system reaches with high probability a metastable regime where a large majority of agents keeps supporting the same opinion for polynomial time. Moreover, this opinion turns out to be the initial majority opinion, whenever the initial bias is slightly larger than its standard deviation. On the contrary, above the threshold, we show that the information about the initial majority opinion is ``lost'' within logarithmic time even when the initial bias is maximum.Interestingly, we show our results have explicit connections to two different concrete frameworks. The first one concerns a specific setting of a well-studied value-sensitive decision mechanism inspired by cross-inhibition in house-hunting honeybee swarms. The second framework consists of a consensus process where a subset of agents behave in a stubborn way.

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