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Graph Foundation Models (GFMs) are emerging as a significant research topic in the graph domain, aiming to develop graph models trained on extensive and diverse data to enhance their applicability across various tasks and domains. Developing GFMs presents unique challenges over traditional Graph Neural Networks (GNNs), which are typically trained from scratch for specific tasks on particular datasets. The primary challenge in constructing GFMs lies in effectively leveraging vast and diverse graph data to achieve positive transfer. Drawing inspiration from existing foundation models in the CV and NLP domains, we propose a novel perspective for the GFM development by advocating for a ``graph vocabulary'', in which the basic transferable units underlying graphs encode the invariance on graphs. We ground the graph vocabulary construction from essential aspects including network analysis, expressiveness, and stability. Such a vocabulary perspective can potentially advance the future GFM design in line with the neural scaling laws. All relevant resources with GFM design can be found here. 

Recent studies on Graph Neural Networks(GNNs) provide both empirical and theoretical evidence supporting their effectiveness in capturing structural patterns on both homophilic and certain heterophilic graphs. Notably, most real-world homophilic and heterophilic graphs are comprised of a mixture of nodes in both homophilic and heterophilic structural patterns, exhibiting a structural disparity. However, the analysis of GNN performance with respect to nodes exhibiting different structural patterns, e.g., homophilic nodes in heterophilic graphs, remains rather limited. In the present study, we provide evidence that Graph Neural Networks(GNNs) on node classification typically perform admirably on homophilic nodes within homophilic graphs and heterophilic nodes within heterophilic graphs while struggling on the opposite node set, exhibiting a performance disparity. We theoretically and empirically identify effects of GNNs on testing nodes exhibiting distinct structural patterns. We then propose a rigorous, non-i.i.d PAC-Bayesian generalization bound for GNNs, revealing reasons for the performance disparity, namely the aggregated feature distance and homophily ratio difference between training and testing nodes. Furthermore, we demonstrate the practical implications of our new findings via (1) elucidating the effectiveness of deeper GNNs; and (2) revealing an over-looked distribution shift factor on graph out-of-distribution problem and proposing a new scenario accordingly.