Edge removal in undirected networks

Strong Converses Are Just Edge Removal Properties

This paper explores the relationship between two ideas in network information theory: edge removal and strong converses. Edge removal properties state that if an edge of small capacity is removed from a network, the capacity region does not change too much. Strong converses state that, for rates outside the capacity region, the probability of error converges to 1 as the blocklength goes to infinity. Various notions of edge removal and strong converse are defined, depending on how edge capacity and error probability scale with blocklength, and relations between them are proved. Each class of st…

A Local Perspective on the Edge Removal Problem

The edge removal problem studies the loss in network coding rates that results when a network communication edge is removed from a given network. It is known, for example, that in networks restricted to linear coding schemes and networks restricted to Abelian group codes, removing an edge e* with capacity Re* reduces the achievable rate on each source by no more than Re*. In this work, we seek to uncover larger families of encoding functions for which the edge removal statement holds. We take a local perspective: instead of requiring that all network encoding functions satisfy certain restrict…

有向图和无向图中的最优函数计算

Optimal Function Computation in Directed and Undirected Graphs

我们认为在传感器网络中的信息聚合的问题，其中一个感兴趣的是在计算的传感器测量的函数。我们允许块处理和研究在有向图和无向图中的网络函数计算。我们研究功能的结构如何影响编码策略，以及交互信息交换的效果。我们首先考虑传感器节点上的有向图G =（V，E），其中目标是确定在收集器节点处实现函数计算的每个边缘上的最佳编码器。我们的目标是表征R^{ |E| }中的速率区域，即，存在具有给定速率的可行编码器的点的集合，该编码器对于渐进大的块长度实现零误差计算。我们确定有向树的解决方案，指定每个边缘的最佳编码器和解码器。对于一般的有向无环图，我们提供了一个外边界的速率区域，找到消歧的要求，为每个切割，并描述的例子，这个外边界是紧的。接下来，我们解决的情况下，节点连接在一个无向树网络，每个节点希望计算一个给定的对称布尔函数的传感器数据。无向边允许交互式计算，因此，我们研究的聚合和通信策略的相互作用的效果。我们专注于总和阈值函数，并确定最坏情况下的最小总位数交换在每个边缘。最佳策略涉及递归的网络内聚合，这让人想起消息传递。在一般图的情况下，我们提出了一个割集下界，和一个可实现的计划的基础上沿树聚集。对于完全图，我们证明了该方案的复杂性不超过最优方案的两倍。

无向图模型选择的联合树框架

A Junction Tree Framework for Undirected Graphical Model Selection

无向图模型是在无向图G* 上定义的联合概率分布，其中图中的顶点索引了随机变量的集合，而边对随机变量之间的条件独立关系进行编码。无向图模型选择（UGMS）问题是在给定无向图模型的观测值的情况下，对图G* 进行估计。本文提出了一个框架，用于将UGMS问题分解为多个子问题的集群和子集的分离器在一个联合树。使用包含G* 中的边的超集的图来构造连接树。我们强调了UGMS使用连接树的三个主要特性。首先，可以使用不同的正则化参数或不同的UGMS算法来学习图的不同部分。这是可能的，因为我们识别的子问题可以相互独立地解决。第二，在一定条件下，基于联合树的UGMS算法可以产生一致的结果与较少的观察比现有算法的通常要求。第三，我们的理论和实验结果都表明，连接树框架在寻找图中的最弱边方面比现有方法做得更好。该性质是第一和第二性质两者的结果。最后，我们注意到，我们的框架是独立的UGMS算法的选择，可以用作包装周围的标准UGMS算法更准确的图估计。

网络编码可在无向多播网络中胜过路由

Network Coding Can Beat Routing in Undirected Multiple-Unicast Networks

本文构造了一个无向多播网络反例，证明网络编码可超越分数路由，并推广至无限网络族，且已在Lean中形式化验证。

Local Conditioning: Exact Message Passing for Cyclic Undirected Distributed Networks

This paper addresses practical implementation of summing out, expanding, and reordering of messages in Local Conditioning (LC) for undirected networks. In particular, incoming messages conditioned on potentially different subsets of the receiving node's relevant set must be expanded to be conditioned on this relevant set, then reordered so that corresponding columns of the conditioned matrices can be fused through element-wise multiplication. An outgoing message is then reduced by summing out loop cutset nodes that are upstream of the outgoing edge. The emphasis on implementation is the primar…