稀疏移动网络中信息传播的紧边界
Tight Bounds on Information Dissemination in Sparse Mobile Networks
摘要：
出于对移动系统日益增长的兴趣，我们研究了在平面上独立移动的代理之间的信息传播动态。形式上，我们认为$k$移动代理执行独立的随机行走的$n$节点网格。在时间$0$，每个代理位于网格的随机节点处，并且一个代理具有谣言。谣言的传播由动态通信图过程${G_t(r) | t \geq 0}$控制，其中两个代理通过$G_t(r)$中的边连接，只要它们在时间$t$处的距离在它们的传输半径$r$内。模拟的物理现实，无线电传输的速度是远远快于运动的代理，我们假设的谣言可以通过连接组件的$G_t$之前，图被改变的运动。我们研究了系统的广播时间$T_B$，这是所有代理知道谣言所花费的时间。我们专注于稀疏的情况下（低于渗流点$r_c \approx \sqrt{n/k}$），在高概率，没有连接组件在$G_t$有超过对数的代理和广播时间是占主导地位的时间，它需要许多独立的随机游走，以满足对方。令人惊讶的是，我们表明，对于一个低于逾渗点的系统的广播时间不依赖于流动性速度和传输半径之间的关系。事实上，我们证明，$T_B = \tilde{O}(n / \sqrt{k})$的任何$0 \leq r < r_c$，即使当传输范围是显着大于在一个步骤中的流动性范围，给出了一个紧密的表征对数因子。我们的结果补充了Peres等人（SODA 2011）最近的结果，Peres等人表明在渗流点以上，广播时间在$k$中是多对数的。
英文摘要：
Motivated by the growing interest in mobile systems, we study the dynamics of information dissemination between agents moving independently on a plane. Formally, we consider $k$ mobile agents performing independent random walks on an $n$-node grid. At time $0$, each agent is located at a random node of the grid and one agent has a rumor. The spread of the rumor is governed by a dynamic communication graph process ${G_t(r) | t \geq 0}$, where two agents are connected by an edge in $G_t(r)$ iff their distance at time $t$ is within their transmission radius $r$. Modeling the physical reality that the speed of radio transmission is much faster than the motion of the agents, we assume that the rumor can travel throughout a connected component of $G_t$ before the graph is altered by the motion. We study the broadcast time $T_B$ of the system, which is the time it takes for all agents to know the rumor. We focus on the sparse case (below the percolation point $r_c \approx \sqrt{n/k}$) where, with high probability, no connected component in $G_t$ has more than a logarithmic number of agents and the broadcast time is dominated by the time it takes for many independent random walks to meet each other. Quite surprisingly, we show that for a system below the percolation point the broadcast time does not depend on the relation between the mobility speed and the transmission radius. In fact, we prove that $T_B = \tilde{O}(n / \sqrt{k})$ for any $0 \leq r < r_c$, even when the transmission range is significantly larger than the mobility range in one step, giving a tight characterization up to logarithmic factors. Our result complements a recent result of Peres et al. (SODA 2011) who showed that above the percolation point the broadcast time is polylogarithmic in $k$.
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