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

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In several real \emph{Multi-Agent Systems} (MAS), it has been observed that only weaker forms of \emph{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) \emph{opinion dynamics} by considering the famous \undecided 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 \emph{metastable consensus} undergoes a \emph{phase transition} for $p=\frac 16$. 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, 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 \emph{stubborn} way.
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Cited literature [43 references]

https://hal.archives-ouvertes.fr/hal-02487650
Contributor : Francesco d'Amore <>
Submitted on : Friday, May 15, 2020 - 10:42:29 AM
Last modification on : Tuesday, May 26, 2020 - 6:50:53 PM

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### Identifiers

• HAL Id : hal-02487650, version 2
• ARXIV : 2005.07423

### Citation

Francesco d'Amore, Andrea Clementi, Emanuele Natale. Phase Transition of a Non-Linear Opinion Dynamics with Noisy Interactions. [Research Report] INRIA Sophia Antipolis - I3S; Università di Roma "Tor Vergata". 2020. ⟨hal-02487650v2⟩

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