Consider the following broadcasting process run on a connected graph $G=(V,E)$. Suppose that $k \ge 2$ agents start on vertices selected from $V$ uniformly and independently at random. One of the agents has a message that she wants to communicate to the other agents. All agents perform independent random walks on $G$, with the message being passed when an agent that knows the message meets an agent that does not know the message. The broadcasting time $\xi(G,k)$ is the time it takes to spread the message to all agents. Our ultimate goal is to gain a better understanding of the broadcasting process run on real-world networks of roads of large cities that might shed some light on the behaviour of future autonomous and connected vehicles. Due to the complexity of road networks, such phenomena have to be studied using simulation in practical applications. In this paper, we study the process on the simplest scenario, i.e., the family of complete graphs, as in this case the problem is analytically tractable. We provide tight bounds for $\xi(K_n,k)$ that hold asymptotically almost surely for the whole range of the parameter $k$. These theoretical results reveal interesting relationships and, at the same time, are also helpful to understand and explain the behaviour we observe in more realistic networks.

中文翻译：

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旅行社模型中的广播时间

考虑以下在连通图 $G=(V,E)$ 上运行的广播过程。假设 $k \ge 2$ 代理从 $V$ 中均匀且独立地随机选择的顶点开始。其中一个代理有一条消息，她想与其他代理通信。所有代理在 $G$ 上执行独立的随机游走，当知道消息的代理遇到不知道消息的代理时，消息被传递。广播时间 $\xi(G,k)$ 是将消息传播到所有代理所需的时间。我们的最终目标是更好地了解在大城市道路的真实世界网络上运行的广播过程，这可能会为未来自动驾驶和联网车辆的行为提供一些启示。由于道路网络复杂，必须在实际应用中使用模拟来研究此类现象。在本文中，我们在最简单的情况下研究该过程，即完全图族，因为在这种情况下，问题在分析上是易于处理的。我们为 $\xi(K_n,k)$ 提供了严格的边界，几乎可以肯定地在参数 $k$ 的整个范围内渐近地保持。这些理论结果揭示了有趣的关系，同时也有助于理解和解释我们在更现实的网络中观察到的行为。