边均匀随机演化图的覆盖时间
Cover Time in Edge-Uniform Stochastically-Evolving Graphs
摘要：
我们定义了随机演化图的一般模型，即\emph{Edge-Uniform Stochastically-Evolving Graphs}。在这个模型中，一个潜在的一般静态图的每一个可能的边缘独立地演化为活的或死的在每个离散的时间步的演化（马尔可夫）随机规则。随机规则对于每个可能的边缘是相同的，并且可以取决于边缘状态的过去的$k \ge 0$观测。我们研究两种随机游走的一个单一的代理发生在这样一个动态图：（i）的\emph{Random Walk with a Delay}（\emph{RWD}），其中在每一步的代理选择（均匀随机）一个事件可能的边缘，即，在底层静态图中的一个事件边，然后它等待直到该边变为活动的来遍历它。（ii）更自然的\emph{Random Walk on what is Available}（\emph{RWA}），其中代理在每个时间步只查看活动的事件边，并且随机均匀地遍历其中之一。我们的研究是对\emph{cover time}的界定，即，每个节点至少被代理访问一次之前的预期时间。对于\emph{RWD}，我们提供了一个第一个上限的情况下$k = 0, 1$相关的\emph{RWD}与一个简单的随机游动的静态图。此外，我们提出了一个修改后的电网络理论捕捉$k = 0$的情况下。对于\emph{RWA}，我们推导出一些第一界的情况下$k = 0$，通过减少\emph{RWA}的\emph{RWD}等效的步行修改延迟。此外，我们还提供了一个框架，它被证明是计算的覆盖时间的一般家庭的随机演化图在指数时间的确切值。最后，我们对边均匀图中的\emph{RWA}的覆盖时间进行了实验，并将实验结果与我们的理论界进行了比较。
英文摘要：
We define a general model of stochastically-evolving graphs, namely the \emph{Edge-Uniform Stochastically-Evolving Graphs}. In this model, each possible edge of an underlying general static graph evolves independently being either alive or dead at each discrete time step of evolution following a (Markovian) stochastic rule. The stochastic rule is identical for each possible edge and may depend on the past $k \ge 0$ observations of the edge's state. We examine two kinds of random walks for a single agent taking place in such a dynamic graph: (i) The \emph{Random Walk with a Delay} (\emph{RWD}), where at each step the agent chooses (uniformly at random) an incident possible edge, i.e., an incident edge in the underlying static graph, and then it waits till the edge becomes alive to traverse it. (ii) The more natural \emph{Random Walk on what is Available} (\emph{RWA}) where the agent only looks at alive incident edges at each time step and traverses one of them uniformly at random. Our study is on bounding the \emph{cover time}, i.e., the expected time until each node is visited at least once by the agent. For \emph{RWD}, we provide a first upper bound for the cases $k = 0, 1$ by correlating \emph{RWD} with a simple random walk on a static graph. Moreover, we present a modified electrical network theory capturing the $k = 0$ case. For \emph{RWA}, we derive some first bounds for the case $k = 0$, by reducing \emph{RWA} to an \emph{RWD}-equivalent walk with a modified delay. Further, we also provide a framework, which is shown to compute the exact value of the cover time for a general family of stochastically-evolving graphs in exponential time. Finally, we conduct experiments on the cover time of \emph{RWA} in Edge-Uniform graphs and compare the experimental findings with our theoretical bounds.
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