马尔可夫网络中离散首过时间概率分布的精确解析表达式

Jaroslav Albert
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引用次数: 0

摘要

首次穿越时间(FPT)是指系统变量首次穿越给定边界所需的时间。在马尔可夫网络中,FPT 是随机行走者从一个节点跳到另一个节点,到达特定节点(目标)所需的时间。如果步行者在每个节点停顿的时间是从连续分布中抽取的,那么 FPT 将是一个连续变量;如果停顿的时间正好是一个单位时间,那么 FPT 将是离散的,等于跳数。我们推导出了马尔可夫网络中离散首次通过时间(DFPT)的精确分析表达式。我们的方法如下:首先,我们将网络中的每条边(两个节点之间的连接)分成 $h$ 单向边,连接 $h$ 虚构节点的级联,并计算连续 FPT(CFPT)。其次,我们将沿边的转换率设为 $h$,并证明随着 $h\to\infty$的增大,原始网络任意两个节点之间的旅行时间分布将趋近于以 1 为中心的三角函数,这相当于停顿持续 1 个单位时间。利用这种方法,我们还计算了 DFPT、目标节点和到达目标节点的节点的联合概率分布。与模拟的比较证实了我们方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Exact analytic expressions for discrete first-passage time probability distributions in Markov networks
The first-passage time (FPT) is the time it takes a system variable to cross a given boundary for the first time. In the context of Markov networks, the FPT is the time a random walker takes to reach a particular node (target) by hopping from one node to another. If the walker pauses at each node for a period of time drawn from a continuous distribution, the FPT will be a continuous variable; if the pauses last exactly one unit of time, the FPT will be discrete and equal to the number of hops. We derive an exact analytical expression for the discrete first-passage time (DFPT) in Markov networks. Our approach is as follows: first, we divide each edge (connection between two nodes) of the network into $h$ unidirectional edges connecting a cascade of $h$ fictitious nodes and compute the continuous FPT (CFPT). Second, we set the transition rates along the edges to $h$, and show that as $h\to\infty$, the distribution of travel times between any two nodes of the original network approaches a delta function centered at 1, which is equivalent to pauses lasting 1 unit of time. Using this approach, we also compute the joint-probability distributions for the DFPT, the target node, and the node from which the target node was reached. A comparison with simulation confirms the validity of our approach.
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