# Differentiable Koopman Operator for Contrastive Learning on Dynamic Graphs

arXiv: https://arxiv.org/abs/2610.02990 · HTML: https://mlsanity.com/paper/2610.02990/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 52 of its 81 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 30 of 52)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Big Self-Supervised Models are Strong Semi-Supervised Learners](https://mlsanity.com/paper/2006.10029/) (2020)
- [Graph Contrastive Learning with Augmentations](https://mlsanity.com/paper/2010.13902/) (2020)
- [Deep Graph Infomax](https://mlsanity.com/paper/1809.10341/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Contrastive Multi-View Representation Learning on Graphs](https://mlsanity.com/paper/2006.05582/) (2020)
- [Deep Graph Contrastive Representation Learning](https://mlsanity.com/paper/2006.04131/) (2020)
- [Inductive Representation Learning on Temporal Graphs](https://mlsanity.com/paper/2002.07962/) (2020)
- [Predicting Dynamic Embedding Trajectory in Temporal Interaction Networks](https://mlsanity.com/paper/1908.01207/) (2019)
- [Inductive Representation Learning in Temporal Networks via Causal Anonymous Walks](https://mlsanity.com/paper/2101.05974/) (2021)
- [Rethinking Graph Neural Networks for Anomaly Detection](https://mlsanity.com/paper/2205.15508/) (2022)
- [Anomaly Detection on Attributed Networks via Contrastive Self-Supervised Learning](https://mlsanity.com/paper/2103.00113/) (2021)
- [Anti-Money Laundering in Bitcoin: Experimenting with Graph Convolutional Networks for Financial Forensics](https://mlsanity.com/paper/1908.02591/) (2019)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Towards Better Dynamic Graph Learning: New Architecture and Unified Library](https://mlsanity.com/paper/2303.13047/) (2023)
- [From Canonical Correlation Analysis to Self-supervised Graph Neural Networks](https://mlsanity.com/paper/2106.12484/) (2021)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Community Interaction and Conflict on the Web](https://mlsanity.com/paper/1803.03697/) (2018)
- [Rethinking and Scaling Up Graph Contrastive Learning: An Extremely Efficient Approach with Group Discrimination](https://mlsanity.com/paper/2206.01535/) (2022)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Anomaly Detection in Dynamic Graphs via Transformer](https://mlsanity.com/paper/2106.09876/) (2021)
- [Graph Anomaly Detection via Multi-Scale Contrastive Learning Networks with Augmented View](https://mlsanity.com/paper/2212.00535/) (2022)
- [Spectral Augmentation for Self-Supervised Learning on Graphs](https://mlsanity.com/paper/2210.00643/) (2022)
- [Simple and Asymmetric Graph Contrastive Learning without Augmentations](https://mlsanity.com/paper/2310.18884/) (2023)

## Abstract (arXiv)

Real-world interaction networks are inherently dynamic: edges form and dissolve as node behavior shifts over time. Most snapshot-based contrastive methods encode temporal dependencies implicitly in encoder weights, without an explicit model of how node representations evolve, making them brittle under distribution shifts. We propose KAIROS (Koopman-Aligned Invariant Representations for Open Dynamic Systems), a self-supervised framework that embeds a differentiable Koopman operator within a dynamic graph contrastive learning loop to linearize temporal evolution in the learned embedding space. A dual-view encoder pairs raw node features with a graph-diffused structural view and is optimized with multi-granularity contrastive objectives across temporal windows. For anomaly detection, KAIROS uses the Koopman prediction residual together with temporal inconsistency and local neighborhood deviation to separate irregular behavior from predictable graph evolution. Evaluated on nine dynamic graph benchmarks, KAIROS achieves state-of-the-art anomaly detection results on all nine datasets, with gains of up to 23.15 ROC-AUC points over prior work, while remaining competitive for unsupervised node classification. These results show that explicit dynamics modeling provides a scalable and effective inductive bias for temporal graph representation learning.