多用户通信网络的最大流最小割定理
Max-Flow Min-Cut Theorems for Multi-User Communication Networks
摘要：
本文提出了四个不同的新思想和结果的通信网络：1）我们表明，中继网络（即通信网络中的不同节点使用相同的编码功能）可以用来模拟动态网络。2)我们介绍{ \em术语模型}，这是一个简单的，无图的符号方法通信网络。3)我们的国家和证明变种的一个定理，关于分散的信息在单接收器通信。4)我们表明，一个抽象的多用户通信问题的可解性是等价的一个合适的中继网络中的单目标通信的可解性。在本文中，我们开发了一些技术上的分歧，这些想法和结果。一个技术性的结果是最大流最小割定理，对于阶数小于1的雷尼熵，假设源是等概率分布的;相反，我们证明了最大流最小割定理对于阶数大于1的雷尼熵是失败的。我们离开的状态定理关于普通香农熵测度（雷尼熵的顺序一和极限情况之间的有效性或失败的定理）作为一个悬而未决的问题。在具有单个接收机的非动态静态通信网络中，门格尔定理的简单应用表明，可以在不适当使用网络编码的情况下实现最佳吞吐量，即仅使用普通的分组交换。这在具有单个接收器的中继网络中明显失败。我们表明，即使是一个强大的方法，如线性网络编码失败悲惨的中继网络。考虑到这一点，值得注意的是，我们相当弱的网络编码形式（使用动态报头的路由）渐近地足以达到容量。
英文摘要：
The paper presents four distinct new ideas and results for communication networks: 1) We show that relay-networks (i.e. communication networks where different nodes use the same coding functions) can be used to model dynamic networks. 2) We introduce {\em the term model}, which is a simple, graph-free symbolic approach to communication networks. 3) We state and prove variants of a theorem concerning the dispersion of information in single-receiver communications. 4) We show that the solvability of an abstract multi-user communication problem is equivalent to the solvability of a single-target communication in a suitable relay network. In the paper, we develop a number of technical ramifications of these ideas and results. One technical result is a max-flow min-cut theorem for the Rényi entropy with order less than one, given that the sources are equiprobably distributed; conversely, we show that the max-flow min-cut theorem fails for the Rényi entropy with order greater than one. We leave the status of the theorem with regards to the ordinary Shannon Entropy measure (Rényi entropy of order one and the limit case between validity or failure of the theorem) as an open question. In non-dynamic static communication networks with a single receiver, a simple application of Menger's theorem shows that the optimal throughput can be achieved without proper use of network coding i.e. just by using ordinary packet-switching. This fails dramatically in relay networks with a single receiver. We show that even a powerful method like linear network coding fails miserably for relay networks. With that in mind, it is noticeable that our rather weak form of network coding (routing with dynamic headers) is asymptotically sufficient to reach capacity.
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