标题： 有限时间内的妥协过程共识 显示英文标题

标题： Finite-time consensus in a compromise process

摘要： 一个妥协过程描述了通过二元相互作用演变的观念。 观念是实数，在每一步中，两个随机选择的代理达成妥协，即他们获得相互作用前观念的平均值。 我们证明，如果代理的数量$N$是2的幂，那么在有限数量的妥协事件后，概率为1地达到共识；否则，如果初始观念处于一般位置，则无法在有限数量的步骤中达到共识。 达到共识所需的步骤数对于$N=2^k$且$k\geq 2$是随机的。 我们证明，通常情况下，达到共识的最小步骤数等于$k\cdot 2^{k-1}$。 对于$N=4$，我们确定了步骤数的分布，例如，我们证明它具有纯粹的指数尾部并计算了所有累积量。 显示英文摘要

摘要： A compromise process describes the evolution of opinions through binary interactions. Opinions are real numbers, and in each step, two randomly selected agents reach a compromise, namely, they acquire the average of their pre-interaction opinions. We prove that if the number $N$ of agents is a power of two, the consensus emerges after a finite number of compromise events with probability one; otherwise, the consensus cannot be reached in a finite number of steps (if initial opinions are in a general position). The number of steps required to reach consensus is random for $N=2^k$ with $k\geq 2$. We prove that typically, the smallest number of steps to reach consensus is equal to $k\cdot 2^{k-1}$. For $N=4$, we determine the distribution of the number of steps, e.g., we show that it has a purely exponential tail and compute all cumulants.