# Discovering hierarchical governing equations for community-structured complex network dynamics with LLM agents

**Authors:** Jinlei Zhang, Zixuan Zhang, Hao Qiu, Feng Chen, Jiateng Yin, Xiaoyong Yan, Xiaolei Ma, Hao Sun, Jianxi Gao, Lixing Yang, Ziyou Gao  
**Lab:** Beijing Jiaotong University · **Published:** 2 Nov 2026  
**Links:** [arXiv](https://arxiv.org/abs/2611.02816) · [PDF](https://arxiv.org/pdf/2611.02816) · [This page](https://omen-bit.github.io/ai-paper-daily/paper/2611.02816.html)

## Abstract

Governing equations provide an interpretable description of the mechanisms underlying complex network dynamics, yet their discovery from observational data remains challenging when shared community-level dynamics coexist with individual-level variations and the relevant functional structures may be absent from the initial candidate library or not explicitly specified in advance. Here we formulate this task as a hierarchical law-discovery problem and introduce SAGE, a large language model-guided self-reflective agent framework for discovering community-shared governing laws and individual-specific dynamical variations. SAGE organizes equation discovery into an iterative Library-Discovery-Review process that first identifies shared community-level structures and then resolves node-specific residual dynamics, while adaptively refining the candidate space to introduce missing dynamical terms when needed. This hierarchical and interpretable formulation enables SAGE to distinguish shared community-level dynamics from individual-level variations and to reveal the mechanisms underlying complex network dynamics. Across two synthetic networks and four real-world networks spanning mobile communication, brain electrophysiology, power grids and urban rail transit, SAGE recovers compact and interpretable governing equations, reproduces observed system dynamics and outperforms representative equation-discovery baselines. The framework remains robust under incomplete observations and can transfer community-level governing laws to nodes with entirely missing observations. These results establish a hierarchical and adaptive approach for disentangling shared and individual dynamical mechanisms in community-structured complex networks.

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