平面图的宏观网络循环
Macroscopic network circulation for planar graphs
摘要：
网络的分析，旨在适当定义的功能，往往侧重于划分成子网络，捕捉所需的功能。相关概念中最主要的是2-划分，它是经典Cheeger不等式的基础，并强调了限制网络各个部分之间可访问性的收缩（瓶颈）。本着同样的精神，本工作的目的是引入一个新的概念，最大全球流通，并探讨3-分区，揭示这种类型的宏观网络的功能。在这里，图循环是由图上的运输网络和概率流（马尔可夫链）激发的。我们的目标是量化网络流量的大规模不平衡，并描绘出调解这种全球特征的关键部分。虽然我们在一般情况下介绍和提出这些概念，但在本文中，我们只研究平面图的情况。我们解释说，标量势可以被确定为封装的概念循环，非常类似的情况下，平面向量场的旋度。除了平面图之外，在一般情况下，确定全球环流的问题目前仍然是一个组合问题。
英文摘要：
The analysis of networks, aimed at suitably defined functionality, often focuses on partitions into subnetworks that capture desired features. Chief among the relevant concepts is a 2-partition, that underlies the classical Cheeger inequality, and highlights a constriction (bottleneck) that limits accessibility between the respective parts of the network. In a similar spirit, the purpose of the present work is to introduce a new concept of maximal global circulation and to explore 3-partitions that expose this type of macroscopic feature of networks. Herein, graph circulation is motivated by transportation networks and probabilistic flows (Markov chains) on graphs. Our goal is to quantify the large-scale imbalance of network flows and delineate key parts that mediate such global features. While we introduce and propose these notions in a general setting, in this paper, we only work out the case of planar graphs. We explain that a scalar potential can be identified to encapsulate the concept of circulation, quite similarly as in the case of the curl of planar vector fields. Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem.
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