Macroscopic network circulation for planar graphs
share
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 concept of macroscopic circulation and to explore 3-partitions that expose this type of feature in flows on networks. It is motivated by transportation networks and probabilistic flows (Markov chains) on graphs. While we introduce and propose such notions in a general setting, we only work out the case of planar graphs. Specifically, we explain that circulation depends on a scalar potential in a manner that is similar to the curl of planar vector fields. For general graphs, assessing global circulation remains at present a combinatorial problem.
READ FULL TEXT