AI 中文总结
该论文研究N个智能体上的异步单调聚合共识动力学，推导了不同类型单调聚合规则下的预期共识时间渐近阶，明确了端点度、独裁者函数等对共识时间的影响。
AI 中文摘要
我们研究N个智能体上的异步共识动力学：每一步，一个均匀选取的智能体将其状态替换为f(Y₁,…,Yᵣ)，其中f是固定的单调聚合规则，Y₁,…,Yᵣ是有放回均匀采样的r个智能体的状态。令T为所有智能体达成一致的首次时间。设f:{0,1}^r→{0,1}是单调且非常数的函数，预期共识时间由两个端点度D₀(f)=#{i:f(eᵢ)=1}和D₁(f)=#{i:f(1−eᵢ)=0}决定，其中eᵢ是第i个标准基向量，1是全1向量。若D₀(f)≠1且D₁(f)≠1，则E[T]=Θ(N(1+log d_f))，其中d_f是到达最近吸引共识前必须改变状态的智能体数量，因此E[T]在所有初始状态下均为O(N log N)；若f是独裁者函数，E[T]由投票者模型公式给出，为Θ(N² Ent(p₀))，其中Ent是二元熵，p₀是状态为1的智能体的初始比例，在所有初始状态下均为绝对常数；否则，D₀(f)和D₁(f)中恰好一个等于1，另一个等于0，此时最坏情况下的预期共识时间为Θ(N^(2−1/m))，其中m≥2是f的残差规则（当D₁(f)=1时为其对偶的残差规则）中强制取值为1所需的最少坐标数。
英文摘要
We study an asynchronous consensus dynamics on $N$ agents: at each step a uniformly chosen agent replaces its state by $f(Y_1,\dots,Y_r)$, where $f$ is a fixed monotone aggregation rule and $Y_1,\dots,Y_r$ are the states of $r$ agents sampled uniformly with replacement. Let $T$ be the first time at which all agents agree. Let $f:\{0,1\}^r\to\{0,1\}$ be monotone and non-constant. The expected consensus time is governed by the two \emph{endpoint degrees} $D_0(f)=\#\{i:f(e_i)=1\}$ and $D_1(f)=\#\{i:f(\mathbf{1}-e_i)=0\}$, where $e_i$ is the $i$th standard basis vector and $\mathbf{1}$ the all-ones vector. If $D_0(f)\ne 1$ and $D_1(f)\ne 1$, then $E[T]=Θ\bigl(N(1+\log d_f)\bigr)$, where $d_f$ counts the agents that must change state before the nearest attracting consensus is reached, so that $E[T]=O(N\log N)$ uniformly over initial states. If $f$ is a dictator, $E[T]$ is given by a voter-model formula and equals $Θ\bigl(N^2 Ent(p_0)\bigr)$ with absolute constants, uniformly over the initial state, where $Ent$ is the binary entropy and $p_0$ the initial fraction of agents in state $1$. Otherwise exactly one of $D_0(f),D_1(f)$ equals $1$ and the other equals $0$. The worst-case expected consensus time is then $Θ(N^{2-1/m})$, where $m\ge 2$ is the least number of coordinates that force the value $1$ in the {\em residual rule} of $f$, defined in the paper (of its dual, when $D_1(f)=1$).
