We study the classical randomized rumour spreading protocol push. Initially, a node in a graph possesses some information, which is then spread in a round based manner. In each round, each informed node chooses uniformly at random one of its neighbours and passes the information to it. The central quantity of interest is the runtime, that is, the number of rounds needed until every node has received the information.

The push protocol and variations of it have been studied extensively. Here we study the case where the underlying graph is complete with $n$ nodes. Even in this most basic setting, specifying the limiting distribution and statistics of it have remained open problems since the protocol was introduced. In our main result we describe the limiting distribution of the runtime. We show that it does not converge, and that it becomes, after the appropriate normalization, asymptotically periodic both on the ${log}_{2}n$ as well as on the $lnn$ scale. Additionally, on suitable subsequences we determine the expected runtime and higher moments of it.

我们研究了经典的随机谣言传播协议推送。最初，图中的节点拥有一些信息，然后以基于回合的方式传播。在每个回合中，每个通知节点随机地均匀选择其邻居之一，并将信息传递给它。感兴趣的中心数量是运行时，即在每个节点接收到信息之前所需的回合数。

所述推送协议和它的变体已被广泛研究。在这里，我们研究基础图完成的情况$ñ$节点。自从引入该协议以来，即使在这种最基本的设置中，指定限制分布和统计仍然是未解决的问题。在我们的主要结果中，我们描述了运行时的限制分布。我们证明它不收敛，并且经过适当的归一化处理后，它渐渐地成为周期渐近的${日志}_{\mathrm{2个}}ñ$ 以及 $lnñ$规模。另外，在合适的子序列上，我们确定预期的运行时间和更高的时刻。