全连接网络中推式流行病数据传播分析
Analysis of Push-type Epidemic Data Dissemination in Fully Connected Networks
摘要：
考虑一个完全连接的节点网络，其中一些节点有一段数据要传播到整个网络。我们分析了以下推式流行病算法：在每一轮推中，每个有数据的节点，即，每个被感染的节点随机选择网络中的$c \in {\mathbb Z}_+$其它节点并发送，即，把数据推给他们。我们把这一轮写为随机游动，它的每一步都对应于随机选择一个被感染的节点;这给出了推轮中新感染节点数量的分布和矩的递归公式。我们使用的分布公式来计算预期的轮数，使一个给定的百分比的网络被感染，并继续在早期的工作中发起的推算法和拉算法（其中易感节点随机选择同行）的数值比较。然后，当网络大小达到$\infty$时，我们推导出随机游动的流体和扩散极限，并推导出推送算法的许多性质：1）在推轮中新感染节点的数量，以及为了使给定百分比的网络被感染所需的随机选择的数量，都是渐近正态的2）对于大型网络，从非零比例的感染节点开始，拉轮平均感染略多的节点3）直到网络的给定比例$λ$被感染的轮数对于几乎所有$λ\in (0,1)$收敛到常数。数值例子的理论结果提供。
英文摘要：
Consider a fully connected network of nodes, some of which have a piece of data to be disseminated to the whole network. We analyze the following push-type epidemic algorithm: in each push round, every node that has the data, i.e., every infected node, randomly chooses $c \in {\mathbb Z}_+$ other nodes in the network and transmits, i.e., pushes, the data to them. We write this round as a random walk whose each step corresponds to a random selection of one of the infected nodes; this gives recursive formulas for the distribution and the moments of the number of newly infected nodes in a push round. We use the formula for the distribution to compute the expected number of rounds so that a given percentage of the network is infected and continue a numerical comparison of the push algorithm and the pull algorithm (where the susceptible nodes randomly choose peers) initiated in an earlier work. We then derive the fluid and diffusion limits of the random walk as the network size goes to $\infty$ and deduce a number of properties of the push algorithm: 1) the number of newly infected nodes in a push round, and the number of random selections needed so that a given percent of the network is infected, are both asymptotically normal 2) for large networks, starting with a nonzero proportion of infected nodes, a pull round infects slightly more nodes on average 3) the number of rounds until a given proportion $λ$ of the network is infected converges to a constant for almost all $λ\in (0,1)$. Numerical examples for theoretical results are provided.
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