# Graph-based Semi-Supervised & Active Learning for Edge Flows

## TL;DR

This paper introduces a graph-based semi-supervised learning method for predicting edge flows on graphs, incorporating flow conservation constraints, and proposes active learning strategies for optimal sensor placement.

## Contribution

It develops a novel SSL framework for edge flows with error bounds and introduces two active learning algorithms tailored for different flow characteristics.

## Key findings

- The method achieves strong performance on synthetic and real-world networks.
- Active learning algorithms effectively select informative edges for flow measurement.
- Flow conservation constraints improve flow prediction accuracy.

## Abstract

We present a graph-based semi-supervised learning (SSL) method for learning edge flows defined on a graph. Specifically, given flow measurements on a subset of edges, we want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flow conservation. These constraints render our approach different from classical graph-based SSL for vertex labels, which posits that tightly connected nodes share similar labels and leverages the graph structure accordingly to extrapolate from a few vertex labels to the unlabeled vertices. We derive bounds for our method's reconstruction error and demonstrate its strong performance on synthetic and real-world flow networks from transportation, physical infrastructure, and the Web. Furthermore, we provide two active learning algorithms for selecting informative edges on which to measure flow, which has applications for optimal sensor deployment. The first strategy selects edges to minimize the reconstruction error bound and works well on flows that are approximately divergence-free. The second approach clusters the graph and selects bottleneck edges that cross cluster-boundaries, which works well on flows with global trends.