In this article, we take a closer look at the reliability of large minimal networks constructed by repeated compositions of the simplest possible networks. For a given number of devices we define the set of all the possible compositions of series and parallel networks of two devices. We then define several partial orders over this set and study their properties. As far as we know the ranking problem has not been addressed before in this context, and this article establishes the first results in this direction. The usual approach when dealing with reliability of two-terminal networks is to determine existence or nonexistence of uniformly most reliable networks. The problem of ranking two-terminal networks is thus more complex, but by restricting our study to the set of compositions we manage to determine and demonstrate the existence of at least a graded poset.

中文翻译：

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火柴棍最小网络的快速可靠性排序

在本文中，我们将仔细研究由最简单的可能网络的重复组合构建的大型最小网络的可靠性。对于给定数量的设备我们定义了两个设备的串联和并联网络的所有可能组合的集合。然后我们在这个集合上定义几个偏序并研究它们的属性。据我们所知，在此背景下之前尚未解决排名问题，本文建立了该方向的第一个结果。处理两端网络可靠性的常用方法是确定一致最可靠网络的存在或不存在。因此，对两端网络进行排序的问题更加复杂，但是通过将我们的研究限制在一组组合物中，我们设法确定并证明了至少一个分级的偏序集的存在。