SPR: A Graph Foundation Model for Transferable Graph Cognition through Spectral Patterns and Relational Geometry

Recently, Graph Foundation Models (GFMs) have attracted increasing attention for their great potential to learn unified and generalizable knowledge across diverse graphs, thereby supporting a wide range of open-ended graph scenarios. However, unlike natural language and images, graphs are less intuitive for humans to understand. Therefore, despite many initial explorations, a key problem remains unresolved: the transferable cognitive mechanism for graphs. Existing GFMs usually adopt intuitively defined mechanisms to uniformly encode transferable graph patterns, such as handcrafted structural templates (e.g., cycles and trees), fixed propagation operators, and predefined random-walk patterns. However, these mechanisms are often prescriptive and rigid, which may limit their ability to simultaneously model diverse graph patterns. This motivates the exploration of a more flexible, graph-native, and transferable cognitive mechanism. To achieve this, we analyze graphs from the spectral perspective, which provides a natural view of graphs, and propose SPR. SPR decomposes graph information into Chebyshev polynomial bases, facilitating the unified cognition of continuously varying graph spectral patterns. We further model cross-graph relational geometry to enhance the transferability of such graph cognition. Extensive experiments across diverse graphs and downstream scenarios demonstrate that SPR achieves more effective and transferable graph cognition.