Phase transition of a non-linear opinion dynamics with noisy interactions: (Extended abstract)

In several realMulti-Agent Systems(MAS), it has been ob-served that only weaker forms ofmetastable consensusare achieved, inwhich a large majority of agents agree on some opinion while other opin-ions continue to be supported by a (small) minority of agents. In thiswork, we take a step towards the investigation of metastable consensusfor complex (non-linear)opinion dynamicsby considering the famousUndecided-Statedynamics in the binary setting, which is known toreach consensus exponentially faster than theVoterdynamics. We pro-pose a simple form of uniform noise in which each message can changeto another one with probabilitypand we prove that the persistence of ametastable consensusundergoes aphase transitionforp=16. In detail,below this threshold, we prove the system reaches with high probabilitya metastable regime where a large majority of agents keeps supportingthe same opinion for polynomial time. Moreover, this opinion turns outto be the initial majority opinion, whenever the initial bias is slightlylarger than its standard deviation. On the contrary, above the thresh-old, we show that the information about the initial majority opinion is“lost” within logarithmic time even when the initial bias is maximum.Interestingly, using a simple coupling argument, we show the equivalencebetween our noisy model above and the model where a subset of agentsbehave in astubbornway.