存在固执智能体时基于多数意见动态的标度律
Scaling Laws for Majority-based Opinion Dynamics in the Presence of Stubborn Agents
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中文总结 AI 辅助
该研究探究多智能体系统中固执智能体对意见动态的影响，基于2k-选择规则，发现达到稳态的时间随固执智能体比例存在尖锐相变，边界区域混合时间为多项式级。
中文摘要 AI 辅助
在多智能体系统中，常常存在坚持特定意见或信念的固执追随者。受此观察启发，本文旨在探究固执智能体如何影响网络中的意见分布，网络中固执智能体与非固执智能体相互作用。为此，我们假设所有智能体的意见属于集合{0,1}，每个非固执智能体依据2k-选择规则更新意见：该智能体均匀随机采样2k个邻居（包括固执和非固执邻居），并采用采样邻居与自身中占多数的意见。我们假设比例为γ_i的智能体是意见i（i∈{0,1}）的固执追随者。人们自然会预期，网络中意见的稳态分布将由固执追随者比例更大的意见主导。我们证明这一预期成立，但达到稳态的时间高度依赖参数γ₀和γ₁的值。当这些参数各自的值及其差值都较小时，达到稳态的时间可能呈指数级（与网络规模相关）增长；相比之下，当γ₀和γ₁中至少一个参数较大时，网络达到稳态的时间仅与网络规模呈对数相关。因此，基于固执智能体的比例，网络动态存在尖锐的相变。我们还刻画了参数γ₀和γ₁处于相变边界时系统的行为：在该边界区域，我们使用Stein方法证明，动态由扩散过程驱动，混合时间为多项式级。
英文摘要
In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$\textit{-choices rule}, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $γ_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $γ_0$ and $γ_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $γ_0$ and $γ_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $γ_0$ and $γ_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.
