Broadcast and consensus are most fundamental tasks in distributed computing. These tasks are particularly challenging in dynamic networks where communication across the network links may be unreliable, e.g., due to mobility or failures. Indeed, over the last years, researchers have derived several impossibility results and high time complexity lower bounds (i.e., linear in the number of nodes $n$) for these tasks, even for oblivious message adversaries where communication networks are rooted trees. However, such deterministic adversarial models may be overly conservative, as many processes in real-world settings are stochastic in nature rather than worst case. This paper initiates the study of broadcast and consensus on stochastic dynamic networks, introducing a randomized oblivious message adversary. Our model is reminiscent of the SI model in epidemics, however, revolving around trees (which renders the analysis harder due to the apparent lack of independence). In particular, we show that if information dissemination occurs along random rooted trees, broadcast and consensus complete fast with high probability, namely in logarithmic time. Our analysis proves the independence of a key variable, which enables a formal understanding of the dissemination process. More formally, for a network with $n$ nodes, we first consider the completely random case where in each round the communication network is chosen uniformly at random among rooted trees. We then introduce the notion of randomized oblivious message adversary, where in each round, an adversary can choose $k$ edges to appear in the communication network, and then a rooted tree is chosen uniformly at random among the set of all rooted trees that include these edges. We show that broadcast completes in $O(k+\log n)$ rounds, and that this it is also the case for consensus as long as $k \le 0.1n$.

翻译：广播和共识是分布式计算中最基础的任务。在动态网络中，由于网络链路可能因移动性、故障等原因而不可靠，这些任务尤其具有挑战性。事实上，近年来研究者已推导出这些任务的若干不可能性结果及高时间复杂度的下界（即与节点数$n$成线性关系），即使对于通信网络为有根树的 oblivious 消息对手也是如此。然而，此类确定性对抗模型可能过于保守，因为现实场景中的许多过程本质上是随机的而非最坏情况。本文首次研究随机动态网络上的广播与共识问题，引入一种随机化的 oblivious 消息对手。我们的模型类似于流行病学中的SI模型，但围绕树结构展开（由于缺乏明显独立性，这使得分析更加困难）。特别地，我们证明若信息传播沿随机有根树进行，则广播和共识能以高概率在对数时间内快速完成。我们的分析证明了一个关键变量的独立性，从而能够从形式上理解传播过程。更正式地，对于包含$n$个节点的网络，我们首先考虑完全随机的情况：每一轮中通信网络从有根树中均匀随机选取。随后我们引入随机化 oblivious 消息对手的概念：每一轮中，对手可选取$k$条边在通信网络中出现，然后从包含这些边的所有有根树集合中均匀随机选取一棵树。我们证明广播可在$O(k+\log n)$轮内完成，且当$k \le 0.1n$时，共识亦能在此时间内达成。