# Importance Ranking in Complex Networks via Influence-aware Causal Node Embedding

arXiv: https://arxiv.org/abs/2511.01228 · HTML: https://mlsanity.com/paper/2511.01228/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 88 references that are in the map.
- First posted on arXiv 3 Nov 2025.

## Builds on (most cited first, 20 of 20)

- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Graph Neural Network-Based Anomaly Detection in Multivariate Time Series](https://mlsanity.com/paper/2106.06947/) (2021)
- [Discovering Invariant Rationales for Graph Neural Networks](https://mlsanity.com/paper/2201.12872/) (2022)
- [Learning Causally Invariant Representations for Out-of-Distribution Generalization on Graphs](https://mlsanity.com/paper/2202.05441/) (2022)
- [Cooperative Game Theory Approaches for Network Partitioning](https://mlsanity.com/paper/1707.03587/) (2017)
- [Debiasing Graph Neural Networks via Learning Disentangled Causal Substructure](https://mlsanity.com/paper/2209.14107/) (2022)
- [A Graph Autoencoder Approach to Causal Structure Learning](https://mlsanity.com/paper/1911.07420/) (2019)
- [End to end learning and optimization on graphs](https://mlsanity.com/paper/1905.13732/) (2019)
- [Estimating Node Importance in Knowledge Graphs Using Graph Neural Networks](https://mlsanity.com/paper/1905.08865/) (2019)
- [Learning causal representations for robust domain adaptation](https://mlsanity.com/paper/2011.06317/) (2020)
- [Continual Causal Inference with Incremental Observational Data](https://mlsanity.com/paper/2303.01775/) (2023)
- [Toward Stronger Robustness of Network Controllability: A Snapback Network Model](https://mlsanity.com/paper/1801.06082/) (2018)
- [Graph Contrastive Invariant Learning from the Causal Perspective](https://mlsanity.com/paper/2401.12564/) (2024)
- [Towards Optimal Robustness of Network Controllability: An Empirical Necessary Condition](https://mlsanity.com/paper/1912.12416/) (2019)
- [Identifying influential nodes in complex networks: Effective distance gravity model](https://mlsanity.com/paper/2011.06348/) (2020)
- [Influencers identification in complex networks through reaction-diffusion dynamics](https://mlsanity.com/paper/1803.01212/) (2018)
- [Measuring and utilizing temporal network dissimilarity](https://mlsanity.com/paper/2111.01334/) (2021)
- [Vital node identification in hypergraphs via gravity model](https://mlsanity.com/paper/2208.04625/) (2022)

## Abstract (arXiv)

Understanding and quantifying node importance is a fundamental problem in network science and engineering, underpinning a wide range of applications such as influence maximization, social recommendation, and network dismantling. Prior research often relies on centrality measures or advanced graph embedding techniques using structural information, followed by downstream classification or regression tasks to identify critical nodes. However, these approaches typically decouple node representation learning from the ranking objective and depend heavily on the topological structure of target networks, leading to feature-task inconsistency and poor cross-network generalization. This paper proposes a novel framework that leverages causal representation learning to obtain robust and invariant node embeddings for cross-network ranking tasks. Specifically, we introduce an influence-aware causal node embedding module within an autoencoder architecture to extract node embeddings that are causally related to node importance. Furthermore, we design a unified optimization framework incorporating a causal ranking loss that jointly optimizes reconstruction and ranking objectives, thereby enabling mutual reinforcement between node representation learning and ranking optimization. This design allows the proposed model to be trained on synthetic networks and to generalize effectively across diverse real-world networks. Extensive experiments on multiple benchmark datasets demonstrate that the proposed model consistently outperforms state-of-the-art baselines in terms of both ranking accuracy and cross-network transferability, offering new insights for network analysis and engineering applications-particularly in scenarios where the target network's structure is inaccessible in advance due to privacy or security constraints.