# MPFlow: Learning Budgeted Max-Flow Optimization on the Lightning Network with Deep Graph Reinforcement Learning

arXiv: https://arxiv.org/abs/2607.08703 · HTML: https://mlsanity.com/paper/2607.08703/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Adversarial robustness and RL in finance, one of 15 directions in Deep reinforcement learning methods.
- Builds on 13 of its 27 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Placed in

- Deep reinforcement learning methods / [Adversarial robustness and RL in finance](https://mlsanity.com/deeprl/adversarial-robustness-and-rl-in-finance/)

## Builds on (most cited first, 13 of 13)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [Neural Algorithmic Reasoning](https://mlsanity.com/paper/2105.02761/) (2021)
- [Flash: Efficient Dynamic Routing for Offchain Networks](https://mlsanity.com/paper/1902.05260/) (2019)
- [Optimally Reliable & Cheap Payment Flows on the Lightning Network](https://mlsanity.com/paper/2107.05322/) (2021)
- [Graph Neural Network Based Node Deployment for Throughput Enhancement](https://mlsanity.com/paper/2209.06905/) (2022)
- [Channel Balance Interpolation in the Lightning Network via Machine Learning](https://mlsanity.com/paper/2405.12087/) (2024)
- [Security and Privacy of Lightning Network Payments with Uncertain Channel Balances](https://mlsanity.com/paper/2103.08576/) (2021)

## Abstract (arXiv)

We address liquidity placement in the Bitcoin Lightning Network (LN): given a fixed budget, which channels should a node open to maximize its routing capacity? We cast this as a budget-constrained combinatorial optimization problem on graphs, selecting k edge additions that maximize s--t max-flow, a theory-grounded measure of routing capacity, and solve it with graph reinforcement learning. Our lightweight agent combines a message-passing policy network with proximal policy optimization (PPO) and action masking, and is trained under a hub-exclusion curriculum: the network's top hubs are removed from training subgraphs, forcing the policy to learn capacity-aware placement rather than hub attachment. In extensive experiments on real Lightning Network snapshots, our method consistently outperforms strong heuristic baselines on the max-flow objective across multiple seeds and unseen graphs. The agent has been deployed in production for peer recommendations, executing 4640 channel-open decisions that cumulatively allocate 267.3 BTC over 16 million across 30 managed nodes.