We study the stabilization time of two common types of influence propagation. In majority processes, nodes in a graph want to switch to the most frequent state in their neighborhood, while in minority processes, nodes want to switch to the least frequent state in their neighborhood. We consider the sequential model of these processes, and assume that every node starts out from a uniform random state. We first show that if nodes change their state for any small improvement in the process, then stabilization can last for up to $\Theta(n^2)$ steps in both cases. Furthermore, we also study the proportional switching case, when nodes only decide to change their state if they are in conflict with a $\frac{1+\lambda}{2}$ fraction of their neighbors, for some parameter $\lambda \in (0,1)$. In this case, we show that if $\lambda < \frac{1}{3}$, then there is a construction where stabilization can indeed last for $\Omega(n^{1+c})$ steps for some constant $c>0$. On the other hand, if $\lambda > \frac{1}{2}$, we prove that the stabilization time of the processes is upper-bounded by $O(n \cdot \log{n})$.
翻译:我们研究两种常见的影响力传播类型的稳定时间。 在多数情况下, 图表中的节点想要切换到其周围最常见的状态, 而在少数民族进程中, 节点想要切换到其周围最不常见的状态。 我们考虑这些过程的顺序模式, 假设每个节点都从一个统一的随机状态开始。 我们首先显示, 如果节点改变其状态, 任何过程的小改进, 那么在两种情况下, 稳定可以持续到$\ Theta (n) 2) 。 此外, 我们还研究比例转换案例, 当节点只决定改变其状态, 如果它们与某个参数$frac{ 1\\\\ lambda} 2} 相冲突, 而他们邻居的一部分 $\ lambda\ in 0, 1, $。 在此情况下, 我们显示, 如果 $lambda < {1\ 3} $, 那么一个建筑可以持续到$\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\