Deep graph models (e.g., graph neural networks and graph transformers) have become important techniques for leveraging knowledge across various types of graphs. Yet, the scaling properties of deep graph models have not been systematically investigated, casting doubt on the feasibility of achieving large graph models through enlarging the model and dataset sizes. In this work, we delve into neural scaling laws on graphs from both model and data perspectives. We first verify the validity of such laws on graphs, establishing formulations to describe the scaling behaviors. For model scaling, we investigate the phenomenon of scaling law collapse and identify overfitting as the potential reason. Moreover, we reveal that the model depth of deep graph models can impact the model scaling behaviors, which differ from observations in other domains such as CV and NLP. For data scaling, we suggest that the number of graphs can not effectively metric the graph data volume in scaling law since the sizes of different graphs are highly irregular. Instead, we reform the data scaling law with the number of edges as the metric to address the irregular graph sizes. We further demonstrate the reformed law offers a unified view of the data scaling behaviors for various fundamental graph tasks including node classification, link prediction, and graph classification. This work provides valuable insights into neural scaling laws on graphs, which can serve as an essential step toward large graph models.

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.

In recent years, there have been remarkable advancements in node classification achieved by Graph Neural Networks (GNNs). However, they necessitate abundant high-quality labels to ensure promising performance. In contrast, Large Language Models (LLMs) exhibit impressive zero-shot proficiency on text-attributed graphs. Yet, they face challenges in efficiently processing structural data and suffer from high inference costs. In light of these observations, this work introduces a label-free node classification on graphs with LLMs pipeline, LLM-GNN. It amalgamates the strengths of both GNNs and LLMs while mitigating their limitations. Specifically, LLMs are leveraged to annotate a small portion of nodes and then GNNs are trained on LLMs' annotations to make predictions for the remaining large portion of nodes. The implementation of LLM-GNN faces a unique challenge: how can we actively select nodes for LLMs to annotate and consequently enhance the GNN training? How can we leverage LLMs to obtain annotations of high quality, representativeness, and diversity, thereby enhancing GNN performance with less cost? To tackle this challenge, we develop an annotation quality heuristic and leverage the confidence scores derived from LLMs to advanced node selection. Comprehensive experimental results validate the effectiveness of LLM-GNN. In particular, LLM-GNN can achieve an accuracy of 74.9% on a vast-scale dataset \products with a cost less than 1 dollar.

Learning on Graphs has attracted immense attention due to its wide real-world applications. The most popular pipeline for learning on graphs with textual node attributes primarily relies on Graph Neural Networks (GNNs), and utilizes shallow text embedding as initial node representations, which has limitations in general knowledge and profound semantic understanding. In recent years, Large Language Models (LLMs) have been proven to possess extensive common knowledge and powerful semantic comprehension abilities that have revolutionized existing workflows to handle text data. In this paper, we aim to explore the potential of LLMs in graph machine learning, especially the node classification task, and investigate two possible pipelines: LLMs-as-Enhancers and LLMs-as-Predictors. The former leverages LLMs to enhance nodes' text attributes with their massive knowledge and then generate predictions through GNNs. The latter attempts to directly employ LLMs as standalone predictors. We conduct comprehensive and systematical studies on these two pipelines under various settings. From comprehensive empirical results, we make original observations and find new insights that open new possibilities and suggest promising directions to leverage LLMs for learning on graphs. Our codes and datasets are available at https://github.com/CurryTang/Graph-LLM.