传染性随机游动
Infectious Random Walks
摘要：
我们研究了$m$移动代理在$n$节点网格上执行独立随机行走的信息（或病毒）传播的动力学。我们制定我们的结果在两个场景：广播和八卦。在广播的情况下，移动代理最初均匀随机放置在网格节点之间。在时间0，一个智能体被告知谣言并开始随机游走。当一个知情的代理人遇到一个不知情的代理人，后者变得知情，并开始一个新的随机游走。我们研究了系统的广播时间，即所有代理知道谣言所需的时间。在八卦场景中，每个智能体在时间0被给予一个不同的谣言，所有智能体开始随机游走。当两个特工见面时，他们会分享他们所知道的所有谣言。我们研究系统的流言传播时间，即所有代理知道所有流言所需的时间。我们证明了广播和八卦时间都是$\tildeΘ(n/\sqrt{m})$ w.h.p.，从而实现了达到对数因子的紧密表征。以前的网格结果提供的边界较弱，只涉及平均时间。在病毒感染的情况下，我们的研究结果的一个推论是，静态和动态移动的代理人感染的速度大致相同。
英文摘要：
We study the dynamics of information (or virus) dissemination by $m$ mobile agents performing independent random walks on an $n$-node grid. We formulate our results in terms of two scenarios: broadcasting and gossiping. In the broadcasting scenario, the mobile agents are initially placed uniformly at random among the grid nodes. At time 0, one agent is informed of a rumor and starts a random walk. When an informed agent meets an uninformed agent, the latter becomes informed and starts a new random walk. We study the broadcasting time of the system, that is, the time it takes for all agents to know the rumor. In the gossiping scenario, each agent is given a distinct rumor at time 0 and all agents start random walks. When two agents meet, they share all rumors they are aware of. We study the gossiping time of the system, that is, the time it takes for all agents to know all rumors. We prove that both the broadcasting and the gossiping times are $\tildeΘ(n/\sqrt{m})$ w.h.p., thus achieving a tight characterization up to logarithmic factors. Previous results for the grid provided bounds which were weaker and only concerned average times. In the context of virus infection, a corollary of our results is that static and dynamically moving agents are infected at about the same speed.
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