We develop a simple and generic method to analyze randomized rumor spreading processes in fully connected networks. In contrast to all previous works, which heavily exploit the precise definition of the process under investigation, we only need to understand the probability and the covariance of the events that uninformed nodes become informed. This universality allows us to easily analyze the classic push, pull, and push-pull protocols both in their pure version and in several variations such as messages failing with constant probability or nodes calling a random number of others each round. Some dynamic models can be analyzed as well, e.g., when the network is a $G(n,p)$ random graph sampled independently each round [Clementi et al. (ESA 2013)]. Despite this generality, our method determines the expected rumor spreading time precisely apart from additive constants, which is more precise than almost all previous works. We also prove tail bounds showing that a deviation from the expectation by more than an additive number of $r$ rounds occurs with probability at most $\exp(-\Omega(r))$. We further use our method to discuss the common assumption that nodes can answer any number of incoming calls. We observe that the restriction that only one call can be answered leads to a significant increase of the runtime of the push-pull protocol. In particular, the double logarithmic end phase of the process now takes logarithmic time. This also increases the message complexity from the asymptotically optimal $\Theta(n \log\log n)$ [Karp, Shenker, Schindelhauer, V\"ocking (FOCS 2000)] to $\Theta(n \log n)$. We propose a simple variation of the push-pull protocol that reverts back to the double logarithmic end phase and thus to the $\Theta(n \log\log n)$ message complexity.

翻译：我们提出了一种简单且通用的方法，用于分析全连接网络中的随机谣言传播过程。与以往所有严重依赖所研究过程精确定义的工作不同，我们仅需理解未知情节点获得信息事件的概率及其协方差。这种通用性使我们能够轻松分析经典推送、拉取及推拉协议，无论是其纯版本还是多种变体（如以恒定概率失败的消息或每轮向随机数量节点发起呼叫）。某些动态模型也可被分析，例如当网络是每轮独立采样的$G(n,p)$随机图时 [Clementi et al. (ESA 2013)]。尽管具有这种通用性，我们的方法能精确确定预期谣言传播时间（仅相差加法常数），这比先前几乎所有工作都更为精确。我们还证明了尾部界：传播时间偏离期望值超过$r$轮加法量的概率最多为$\exp(-\Omega(r))$。我们进一步利用该方法讨论了节点可应答任意数量来电的常见假设，并发现当限制只能应答一个来电时，推拉协议运行时间显著增加：特别是双对数结束阶段现在需要对数时间，同时消息复杂度从渐近最优的$\Theta(n \log\log n)$ [Karp, Shenker, Schindelhauer, V\"ocking (FOCS 2000)] 增加至$\Theta(n \log n)$。我们提出了一种简单的推拉协议变体，使其恢复双对数结束阶段，从而将消息复杂度降回$\Theta(n \log\log n)$。