随机正则图与扩张图上的谣言传播
Rumor Spreading on Random Regular Graphs and Expanders
摘要：
广播算法是分布式系统的重要组成部分。在这项工作中，我们调查的经典和研究推模型的典型性能。假设一个给定网络中的一个节点最初持有一些信息。在每一轮中，每个被通知的节点都独立地随机选择一个邻居，并向其发送消息。在本文中，我们考虑了随机网络，其中每个顶点的度至少为3，即，从具有n个顶点的所有d-正则图的集合中随机均匀地绘制基础图。我们证明了Push模型以概率1 - o（1）将消息广播到（1 + o（1））C_d ln n轮内的所有节点，其中C_d = 1/ ln（2（1-1/d））- 1/（d ln（1 - 1/d））。特别是，我们可以精确地描述节点度对推送模型的典型广播时间的影响。此外，我们考虑伪随机正则网络，其中我们假设每个节点的度非常大。我们证明了广播时间是（1+o（1））C ln n，概率为1 - o（1），其中C= 1/ ln 2 + 1，是C d随d增长的极限。
英文摘要：
Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.
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