Stochastic Analysis of Rumor Spreading with Multiple Pull Operations

Abstract

We propose and analyze a new asynchronous rumor spreading protocol to deliver a rumor to all the nodes of a large-scale distributed network. This spreading protocol relies on what we call a k-pull operation, with $k \ge 2$. Specifically a k-pull operation consists, for an uninformed node s, in contacting $k-1$ other nodes at random in the network, and if at least one of them knows the rumor, then node s learns it. We perform a thorough study of the total number $T_{k,n}$ of k-pull operations needed for all the n nodes to learn the rumor. We compute the expected value and the variance of $T_{k,n}$, together with their limiting values when n tends to infinity. We also analyze the limiting distribution of $(T_{k,n} - {E}(T_{k,n}))/n$ and prove that it has a double exponential distribution when n tends to infinity. Finally, we show that when $k > 2$, our new protocol requires less operations than the traditional 2-push-pull and 2-push protocols by using stochastic dominance arguments. All these results generalize the standard case $k=2$.

Appendix

Let $X_1, \ldots ,X_n$ be n independent geometric random variables with possibly distinct parameters, i.e. such that $X_i \sim \mathcal {G}(p_i)$ with $p_i \in (0,1]$. Let $X = X_1 + \cdots + X_n$, $\mu = {E}(X)$ and $p_{*} = \min _{i=1,\ldots ,n} p_i$. We then have the following results which have been proved in Janson (2018).

Theorem 13

For any $p_1,\ldots ,p_n \in (0,1]$ and any $\lambda \ge 1$,

$$\begin{aligned} {P}\{X \ge \lambda \mu \} \le e^{-p_*\mu (\lambda - 1 - \ln (\lambda ))}. \end{aligned}$$

Theorem 14

For any $p_1,\ldots ,p_n \in (0,1]$ and any $\lambda \le 1$,

$$\begin{aligned} {P}\{X \le \lambda \mu \} \le e^{-p_*\mu (\lambda - 1 - \ln (\lambda ))}. \end{aligned}$$