State Space Models over Directed Graphs

Directed graphs are ubiquitous across numerous domains, where the directionality of edges encodes critical causal dependencies. However, existing GNNs and graph Transformers tailored for directed graphs face two major challenges: (1) effectively capturing long-range causal dependencies derived from directed edges; (2) balancing accuracy and training efficiency when processing large-scale graph datasets. In recent years, state space models (SSMs) have achieved substantial progress in causal sequence tasks, and their variants designed for graphs have demonstrated state-of-the-art accuracy while maintaining high efficiency across various graph learning benchmarks. However, existing graph state space models are exclusively designed for undirected graphs, which limits their performance in directed graph learning. To this end, we propose an innovative approach DirEgo2Token which sequentializes directed graphs via k-hop ego graphs. This marks the first systematic extension of state space models to the field of directed graph learning. Building upon this, we develop DirGraphSSM, a novel directed graph neural network architecture that implements state space models on directed graphs via the message-passing mechanism. Experimental results demonstrate that DirGraphSSM achieves state-of-the-art performance on three representative directed graph learning tasks while attaining competitive performance on two additional tasks with 1.5$\times $ to 2$\times $ training speed improvements compared to existing state-of-the-art models.