Phase Transitions in Biased Opinion Dynamics with 2-choices Rule
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910427677458432 |
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| author | Mukhopadhyay, Arpan |
| author_facet | Mukhopadhyay, Arpan |
| contents | We consider a model of binary opinion dynamics where one opinion is inherently 'superior' than the other and social agents exhibit a 'bias' towards 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 the opinions of the other agents. With probability $1-α$ it adopts the majority opinion among two randomly sampled neighbours and itself. We are interested in the time it takes for the network to converge to a consensus state where all the agents adopt the superior alternative. In a fully connected network of size $n$, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $Θ(n \log n)$ when the bias parameter $α$ is sufficiently high, i.e., $α> α_c$ where $α_c$ is a threshold parameter that is uniquely characterised. When the bias is low, i.e., when $α\in (0,α_c]$, 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 $p_c(α)$. If this is not the case, then we show that the network takes $Ω(\exp(Θ(n)))$ time on average to reach consensus. Through numerical simulations we observe similar behaviour for other classes of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_12245 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Phase Transitions in Biased Opinion Dynamics with 2-choices Rule Mukhopadhyay, Arpan Probability 60J20 (Primary) 91D30, 82C22 (Secondary) We consider a model of binary opinion dynamics where one opinion is inherently 'superior' than the other and social agents exhibit a 'bias' towards 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 the opinions of the other agents. With probability $1-α$ it adopts the majority opinion among two randomly sampled neighbours and itself. We are interested in the time it takes for the network to converge to a consensus state where all the agents adopt the superior alternative. In a fully connected network of size $n$, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $Θ(n \log n)$ when the bias parameter $α$ is sufficiently high, i.e., $α> α_c$ where $α_c$ is a threshold parameter that is uniquely characterised. When the bias is low, i.e., when $α\in (0,α_c]$, 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 $p_c(α)$. If this is not the case, then we show that the network takes $Ω(\exp(Θ(n)))$ time on average to reach consensus. Through numerical simulations we observe similar behaviour for other classes of graphs. |
| title | Phase Transitions in Biased Opinion Dynamics with 2-choices Rule |
| topic | Probability 60J20 (Primary) 91D30, 82C22 (Secondary) |
| url | https://arxiv.org/abs/2208.12245 |