# Efficient quantification on large-scale networks

arXiv: https://arxiv.org/abs/2503.15267 · HTML: https://mlsanity.com/paper/2503.15267/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 60 references that are in the map.
- First posted on arXiv 19 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [A Fair Comparison of Graph Neural Networks for Graph Classification](https://mlsanity.com/paper/1912.09893/) (2019)
- [Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2102.06462/) (2021)
- [A Gentle Introduction to Deep Learning for Graphs](https://mlsanity.com/paper/1912.12693/) (2019)
- [Time for a change: a tutorial for comparing multiple classifiers through Bayesian analysis](https://mlsanity.com/paper/1606.04316/) (2016)
- [Fast and Deep Graph Neural Networks](https://mlsanity.com/paper/1911.08941/) (2019)
- [Fisher consistency for prior probability shift](https://mlsanity.com/paper/1701.05512/) (2017)
- [Are Graph Neural Networks Miscalibrated?](https://mlsanity.com/paper/1905.02296/) (2019)
- [Evaluation Measures for Quantification: An Axiomatic Approach](https://mlsanity.com/paper/1809.01991/) (2018)
- [Measuring Fairness Under Unawareness of Sensitive Attributes: A Quantification-Based Approach](https://mlsanity.com/paper/2109.08549/) (2021)
- [Binary Quantification and Dataset Shift: An Experimental Investigation](https://mlsanity.com/paper/2310.04565/) (2023)
- [Influence of augmented humans in online interactions during voting events](https://mlsanity.com/paper/1803.08086/) (2018)
- [Addressing Heterophily in Node Classification with Graph Echo State Networks](https://mlsanity.com/paper/2305.08233/) (2023)

## Built on by (most cited first, 1 of 1)

- [Geometry-Aware Bayesian Quantification via Compositional Data Analysis](https://mlsanity.com/paper/2607.04977/) (2026)

## Abstract (arXiv)

Network quantification (NQ) is the problem of estimating the proportions of nodes belonging to each class in subsets of unlabelled graph nodes. When prior probability shift is at play, this task cannot be effectively addressed by first classifying the nodes and then counting the class predictions. In addition, unlike non-relational quantification, NQ demands enhanced flexibility in order to capture a broad range of connectivity patterns, resilience to the challenge of heterophily, and scalability to large networks. In order to meet these stringent requirements, we introduce XNQ, a novel method that synergizes the flexibility and efficiency of the unsupervised node embeddings computed by randomized recursive Graph Neural Networks, with an Expectation-Maximization algorithm that provides a robust quantification-aware adjustment to the output probabilities of a calibrated node classifier. In an extensive evaluation, in which we also validate the design choices underpinning XNQ through comprehensive ablation experiments, we find that XNQ consistently and significantly improves on the best network quantification methods to date, thereby setting the new state of the art for this challenging task. XNQ also provides a training speed-up of up to 10x-100x over other methods based on graph learning.