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

Francesco d'Amore 1 Andrea Clementi 2 Emanuele Natale 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Consensus processes play a key role in self-organizing phenomena observed in several real Multi-Agent Systems (MAS). In models proposed for social and biological systems, consensus has been thoroughly investigated in the context of opinion dynamics. In such scenarios, it has been nevertheless 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) dynamics by considering the famous Undecided-State dynamics, which is known to reach consensus exponentially faster than the Voter dynamics by making use of only one single additional agent state. We consider 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 = 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 majority opinion in the initial configuration, whenever the initial bias is slightly larger than its standard deviation (the latter being computed under the random uniform configuration). 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 (i.e., even when the system starts from any full-consensus configuration). Interestingly, using a simple coupling argument, we show the equivalence between our noisy model above and the model where a subset of agents behave in a stubborn way. This implies a similar phase transition in the latter model as well.
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Francesco d'Amore, Andrea Clementi, Emanuele Natale. Phase Transition of a Non-Linear Opinion Dynamics with Noisy Interactions. [Research Report] INRIA Sophia Antipolis - I3S. 2020. ⟨hal-02487650⟩

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