Macroscopic Network Circulation for Planar Graphs

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; this underlies the classical Cheeger inequality and highlights a “constriction” (bottleneck) that limits accessibility between the respective parts of the network. In a similar spirit, we explore a notion of
global circulation
which necessitates a concept of a 3-partition that exposes this macroscopic feature of network flows. Graph circulation is often present in transportation networks as well as in certain biological networks. We introduce a notion of circulation for general graphs and then focus on planar graphs. For the case of the latter, we explain that a scalar potential characterizes circulation in complete analogy with the curl of planar vector fields and we present an algorithm for determining values of the potential and, hence, quantify circulation. We then discuss alternative notions of circulation, explain how these may depend on graph embedding, draw parallels between networks and Helmholtz–Hodge decomposition of vector fields, and conclude with a suggestive application of the framework in detecting abnormalities in cardiac circulatory physiology.