# Learning Graphs through Continuous Information Entropy Fields

arXiv: https://arxiv.org/abs/2606.22895 · HTML: https://mlsanity.com/paper/2606.22895/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 26 of 26)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [InfoGraph: Unsupervised and Semi-supervised Graph-Level Representation Learning via Mutual Information Maximization](https://mlsanity.com/paper/1908.01000/) (2019)
- [Graph Random Neural Network for Semi-Supervised Learning on Graphs](https://mlsanity.com/paper/2005.11079/) (2020)
- [Graph Neural Networks with convolutional ARMA filters](https://mlsanity.com/paper/1901.01343/) (2019)
- [Finding Global Homophily in Graph Neural Networks When Meeting Heterophily](https://mlsanity.com/paper/2205.07308/) (2022)
- [Graph Information Bottleneck](https://mlsanity.com/paper/2010.12811/) (2020)
- [SGFormer: Simplifying and Empowering Transformers for Large-Graph Representations](https://mlsanity.com/paper/2306.10759/) (2023)
- [Generative Causal Explanations for Graph Neural Networks](https://mlsanity.com/paper/2104.06643/) (2021)
- [Graph Structure Learning with Variational Information Bottleneck](https://mlsanity.com/paper/2112.08903/) (2021)
- [AI Flow at the Network Edge](https://mlsanity.com/paper/2411.12469/) (2024)
- [Relating Graph Neural Networks to Structural Causal Models](https://mlsanity.com/paper/2109.04173/) (2021)
- [Positive-incentive Noise](https://mlsanity.com/paper/2212.09541/) (2022)
- [Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs](https://mlsanity.com/paper/2406.02997/) (2024)
- [Energy-based Epistemic Uncertainty for Graph Neural Networks](https://mlsanity.com/paper/2406.04043/) (2024)
- [Bounded and Uniform Energy-based Out-of-distribution Detection for Graphs](https://mlsanity.com/paper/2504.13429/) (2025)
- [Commute Graph Neural Networks](https://mlsanity.com/paper/2407.01635/) (2024)
- [Enhancing the Influence of Labels on Unlabeled Nodes in Graph Convolutional Networks](https://mlsanity.com/paper/2411.02279/) (2024)
- [Predicting and Interpreting Energy Barriers of Metallic Glasses with Graph Neural Networks](https://mlsanity.com/paper/2401.08627/) (2023)

## Abstract (arXiv)

Graph theory is inherently descriptive, capturing what relationships exist but not why they arise, because it treats edges as primitive constructs. This paper proposes a new explanatory framework for graph learning, where relationships emerge from latent continuous information entropy fields, and a graph becomes a discrete instantiation of an underlying field. To formalize this field, we introduce the Field-informed Graph Network (FGN). It learns a scalar field from node features and leverages it to modulate message passing. The information-theoretic objective balances structural fidelity with field smoothness, forming a self-reinforcing loop. In this loop, the field modulates information diffusion through field-modulated weighting, and the updated node representations iteratively refine the field. As a result, FGN learns by simulating its own co-evolution. Extensive experiments on node classification and graph classification benchmarks demonstrate superior performance, robustness to perturbations, and structurally coherent field representations.