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

Abstract--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 methods typically decouple node representation learning from the ranking objective and rely on the topological structure of target networks, leading to feature-task inconsistency and limited generalization across networks. This paper proposes a novel framework that leverages causal representation learning to get robust, invariant node embeddings for cross-network ranking tasks. Firstly, we introduce an influence-aware causal node embedding module within an autoencoder architecture to extract node embeddings that are causally related to node importance. Moreover, we introduce a causal ranking loss and design a unified optimization framework that jointly optimizes the reconstruction and ranking objectives, 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. Complex networks provide a powerful framework for modeling and analyzing a wide range of systems across diverse domains, including social networks, transportation systems, and biological networks [1]. In these networks, nodes represent entities within a real system such as individuals, infrastructure components, or functional units, while edges capture interactions or relationships between them. A key challenge in network science and engineering is identifying important nodes, as they play pivotal roles in maintaining network functionality, performance, stability, and robustness [2].