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

d'Amore, Francesco; Clementi, Andrea E. F.; Natale, Emanuele

doi:10.1007/978-3-030-54921-3_15

Cited by 5 publications

(13 citation statements)

References 35 publications

(78 reference statements)

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“…We consider the dynamics in the binary opinion case over the fully connected network. We introduce in the process an uniform communication noise feature, following the definition in [17] and for which we give an equivalent formulation: for each communication with a sampled neighbor, there is probability p ∈ (0, 1) that it is noisy, i.e. the received opinion is sampled u.a.r.…”

Section: Our Results and Their Consequencesmentioning

confidence: 99%

“…Instead, with probability 1 − p the communication is unaffected by noise. As shown in [17], this noise model (over the complete network) is equivalent to a model without any communication noise and where two communities of stubborn agents (that is, they never change opinion) of equal size pn 2(1−p) are present in the network, where each of the two community holds a different opinion. Even though the complete graph is a strong assumption for such communication networks, we remark that, at every round, an agent pulls an opinion from three neighbors: therefore, the round-per-round communication pattern results is a dynamic graph with O (n) edges.…”

Section: Our Results and Their Consequencesmentioning

confidence: 99%

“…To the best of our knowledge, the Undecided-State dynamics is the first non-linear opinion dynamics analyzed in the presence of uniform communication noise [17]. The aforementioned dynamics exhibits a phase-transition which depends on the noise parameter, and a metastable phase of almost-consensus is quickly reached and kept for long time when the noise isn't too high.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, they involve sending complicated codes through communication links, and it is reasonable to assume that biological type entities communicate between each other in a simpler way. For this reason, in recent years many works have been focusing on the study of the (majority) consensus problem where the communication between entities is unreliable and subjected to uniform noise [16,17,23,24].…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we consider the popular 3-Majority dynamics, which is based on majority update-rules, the latter being widely employed also in the biological research field [13,20]. In particular, we introduce in the system an uniform communication noise feature, following the definition of [17]. It has been proven that such dynamics, without communication noise, has a very similar behaviour to that of the Undecided-State dynamics [6].…”

Section: Introductionmentioning

confidence: 99%

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

d'Amore

Ziccardi²

2021

Preprint

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Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. In particular, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics that are not strongly affected by noisy communications. In this work, we study the popular 3-Majority dynamics, an opinion dynamics which has been proved to be an efficient protocol for the majority consensus problem, in which we introduce a simple feature of uniform communication noise, following (d'Amore et al. 2020). We prove that in the fully connected communication network of n agents and in the binary opinion case, the process induced by the 3-Majority dynamics exhibits a phase transition. For a noise probability p < 1/3, the dynamics reaches in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. Furthermore, departing from previous analyses, we further characterize this phase by showing that there exists an attractive equilibrium value s eq ∈ [n] for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, the agreement opinion turns out to be the initial majority one if the bias towards it is of magnitude Ω √ n log n in the initial configuration. If, instead, p > 1/3, no form of consensus is possible, and any information regarding the initial majority opinion is lost in logarithmic time with high probability. Despite more communications per-round are allowed, the 3-Majority dynamics surprisingly turns out to be less resilient to noise than the Undecided-State dynamics (d'Amore et al. 2020), whose noise threshold value is p = 1/2.

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Section: Our Results and Their Consequencesmentioning

confidence: 99%

Section: Our Results and Their Consequencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

d'Amore

Ziccardi²

2021

Preprint

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

d'Amore

Ziccardi

2022

Structural Information and Communication Complexity

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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 Non-linear Opinion Dynamics with Noisy Interactions

Cited by 5 publications

References 35 publications

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

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

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

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

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