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：广播与共识是分布式计算中最基本的任务。在动态网络中，由于移动性或故障等原因，跨网络链路的通信可能不可靠，这些任务尤其具有挑战性。事实上，在过去几年中，研究人员已经针对这些任务推导出了若干不可能性结果和高时间复杂度的下界（即与节点数 $n$ 呈线性关系），即使对于通信网络为有根树的遗忘消息对抗模型也是如此。然而，这种确定性对抗模型可能过于保守，因为现实世界中的许多过程本质上是随机的，而非最坏情况。本文首次研究了随机动态网络上的广播与共识问题，引入了随机化遗忘消息对抗模型。我们的模型让人联想到流行病学中的SI模型，但围绕树展开（由于明显缺乏独立性，这使得分析更加困难）。具体而言，我们证明，如果信息沿着随机有根树传播，广播和共识极有可能快速完成，即在对数时间内完成。我们的分析证明了一个关键变量的独立性，从而能够形式化地理解传播过程。更形式化地说，对于一个具有 $n$ 个节点的网络，我们首先考虑完全随机的情况，即每一轮通信网络都从有根树中均匀随机选择。然后，我们引入随机化遗忘消息对抗的概念，其中在每一轮中，对抗者可以选择 $k$ 条边出现在通信网络中，然后从包含这些边的所有有根树集合中均匀随机选择一个有根树。我们证明广播在 $O(k+\log n)$ 轮内完成，并且只要 $k \le 0.1n$，共识也是如此。