In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures.We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand, pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary may modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down.On the technical side, we develop two novel methods for the analysis of randomized rumour-spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemerédi’s Regularity Lemma.

中文翻译：

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随机谣言传播的鲁棒性

在这项工作中，我们考虑了三个经过充分研究的广播协议：推,拉和推拉. 所有这些模型的一个关键特性，也是它们受欢迎的一个重要原因，是它们被认为是非常健壮的，因为它们简单、随机，而且至关重要的是，它们没有明确利用底层图的全局结构。虽然存在零星的结果，但没有系统的理论处理来量化这些模型的稳健性。在这里，我们从两个正交方面研究这个问题：（对抗性）基础图的修改和消息传输失败。我们特别探讨了以下概念地方弹性：从图开始，我们调查对手可以在每个顶点删除的边的哪一部分，因此协议需要更多的轮次来广播信息。我们的主要发现建立了三个模型之间的分离。一方面，拉对我们考虑的所有参数都是稳健的。另一方面，推可能会显着减慢速度，即使对手可能仅通过任意小的正数来修改顶点的度数。最后，推拉在不考虑消息传输失败的情况下是稳健的，否则可能会减慢速度。在技术方面，我们开发了两种用于分析随机谣言传播协议的新方法。首先，我们利用自边界函数的概念来显着促进基于轮的分析：我们表明，对于任何图，知情顶点的增长方差都受其期望的限制，因此集中结果立即出现。其次，为了控制图的对抗性修改，我们使用了极值图论中的一个强大工具，即 Szemerédi 的正则引理。