Phase Transition of the 3-Majority Dynamics with Uniform Communication Noise

d'Amore, Francesco; Ziccardi, Isabella

doi:10.1007/978-3-031-09993-9_6

Cited by 5 publications

(1 citation statement)

References 40 publications

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“…This is unlike the previous papers [1,2], where the dynamics is studied as a function of only with a fixed value of p (specifically, p = 0). We show that phase transitions can occur with respect to both and p. Similar phase transitions have been recently reported in [8,9,11] for noisy versions of the k-majority dynamics. However, the focus of these papers is a state of "near consensus" where both opinions coexist, but the proportion of agents with one of the opinions is arbitrarily small.…”

Section: Related Literaturesupporting

confidence: 86%

Phase transitions in biased opinion dynamics with 2-choices rule

Mukhopadhyay

2023

Prob. Eng. Inf. Sci.

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We consider a model of binary opinion dynamics where one opinion is inherently “superior” than the other, and social agents exhibit a “bias” toward the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability α > 0 irrespective of its current opinion and opinions of other agents. With probability $1-\alpha$ , it adopts majority opinion among two randomly sampled neighbors and itself. We are interested in the time it takes for the network to converge to a consensus on the superior alternative. In a complete graph of size n, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $\Theta(n\,\log n)$ when the bias parameter α is sufficiently high, that is, $\alpha \gt \alpha_c$ where α c is a threshold parameter that is uniquely characterized. When the bias is low, that is, when $\alpha \in (0,\alpha_c]$ , we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(\alpha)$ . If this is not the case, then we show that the network takes $\Omega(\exp(\Theta(n)))$ time on average to reach consensus.

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Section: Related Literaturesupporting

confidence: 86%

Phase transitions in biased opinion dynamics with 2-choices rule

Mukhopadhyay

2023

Prob. Eng. Inf. Sci.

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Phase transition of the k-majority dynamics in biased communication models

et al. 2023

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Consider a graph where each of the n nodes is either in state $$\mathcal {R}$$ R or $$\mathcal {B}$$ B . Herein, we analyze the synchronousk-Majoritydynamics, where in each discrete-time round nodes simultaneously sample k neighbors uniformly at random with replacement and adopt the majority state among those of the nodes in the sample (breaking ties uniformly at random). Differently from previous work, we study the robustness of the k-Majority in maintaining a$$\mathcal {R}$$ R majority, when the dynamics is subject to two forms of bias toward state $$\mathcal {B}$$ B . The bias models an external agent that attempts to subvert the initial majority by altering the communication between nodes, with a probability of success p in each round: in the first form of bias, the agent tries to alter the communication links by transmitting state $$\mathcal {B}$$ B ; in the second form of bias, the agent tries to corrupt nodes directly by making them update to $$\mathcal {B}$$ B . Our main result shows a sharp phase transition in both forms of bias. By considering initial configurations in which every node has probability $$q \in (\frac{1}{2},1]$$ q ∈ ( 1 2 , 1 ] of being in state $$\mathcal {R}$$ R , we prove that for every $$k\ge 3$$ k ≥ 3 there exists a critical value $$p_{k,q}^\star $$ p k , q ⋆ such that, with high probability, the external agent is able to subvert the initial majority either in $$n^{\omega (1)}$$ n ω ( 1 ) rounds, if $$p<p_{k,q}^\star $$ p < p k , q ⋆ , or in O(1) rounds, if $$p>p_{k,q}^\star $$ p > p k , q ⋆ . When $$k<3$$ k < 3 , instead, no phase transition phenomenon is observed and the disruption happens in O(1) rounds for $$p>0$$ p > 0 .

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Phase transition of a nonlinear opinion dynamics with noisy interactions

2022

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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 famous 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 = 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, 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.

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Phase Transition of the 3-Majority Dynamics with Uniform Communication Noise

Cited by 5 publications

References 40 publications

Phase transitions in biased opinion dynamics with 2-choices rule

Phase transitions in biased opinion dynamics with 2-choices rule

Phase transition of the k-majority dynamics in biased communication models

Phase transition of a nonlinear opinion dynamics with noisy interactions

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