Graph neural networks (GNNs) have been successful in learning representations from graphs. Many popular GNNs follow the pattern of aggregate-transform: they aggregate the neighbors' attributes and then transform the results of aggregation with a learnable function. Analyses of these GNNs explain which pairs of non-identical graphs have different representations. However, we still lack an understanding of how similar these representations will be. We adopt kernel distance and propose transform-sum-cat as an alternative to aggregate-transform to reflect the continuous similarity between the node neighborhoods in the neighborhood aggregation. The idea leads to a simple and efficient graph similarity, which we name Weisfeiler-Leman similarity (WLS). In contrast to existing graph kernels, WLS is easy to implement with common deep learning frameworks. In graph classification experiments, transform-sum-cat significantly outperforms other neighborhood aggregation methods from popular GNN models. We also develop a simple and fast GNN model based on transform-sum-cat, which obtains, in comparison with widely used GNN models, (1) a higher accuracy in node classification, (2) a lower absolute error in graph regression, and (3) greater stability in adversarial training of graph generation.

Pseudo labeling (PL) is a wide-applied strategy to enlarge the labeled dataset by self-annotating the potential samples during the training process. Several works have shown that it can improve the graph learning model performance in general. However, we notice that the incorrect labels can be fatal to the graph training process. Inappropriate PL may result in the performance degrading, especially on graph data where the noise can propagate. Surprisingly, the corresponding error is seldom theoretically analyzed in the literature.

S1.1 Step-by-step derivation of min-max optimization in Section 2.2.1 By substituting Eq. 2 into Eq. 1 in the main manuscript, we can obtain the objective function of subscript z (we temporarily drop ifor clarity): J(z) = max Since z might be in high dimensional space, solving such a large system of linear equations under the constraint |z| 1is oftentimes computationally challenging. In order to find a practical solution for z that satisfies the constrained minimization problem in Eq. By setting zl as point of coincidence, we can find a separable majorizer of M(z) by adding the non-negative function (z zl) (βI Gx Gx)(z zl) (S6) 37th Conference on Neural Information Processing Systems (NeurIPS 2023). Note, to unify the format, we use the matrix transpose property in Eq. Then, the next step is to find z RN that minimizes z z 2bz subject to the constraint |z| 1. Let's first consider the simplest case where z is a scalar: argmin If b 1, then the solution is z = b.

Zero-shot node classification is a vital task in the field of graph data processing, aiming to identify nodes of classes unseen during the training process. Prediction bias is one of the primary challenges in zero-shot node classification, referring to the model's propensity to misclassify nodes of unseen classes as seen classes. However, most methods introduce external knowledge to mitigate the bias, inadequately leveraging the inherent cluster information within the unlabeled nodes. To address this issue, we employ spectral analysis coupled with learnable class prototypes to discover the implicit cluster structures within the graph, providing a more comprehensive understanding of classes. In this paper, we propose a spectral approach for zero-shot node classification (SpeAr). Specifically, we establish an approximate relationship between minimizing the spectral contrastive loss and performing spectral decomposition on the graph, thereby enabling effective node characterization through loss minimization. Subsequently, the class prototypes are iteratively refined based on the learned node representations, initialized with the semantic vectors. Finally, extensive experiments verify the effectiveness of the SpeAr, which can further alleviate the bias problem.

Referring to Section 4.3, FedSage+ includes two phases. Firstly, all data owners in the distributed subgraph system jointly train NeighGen models through sharing gradients. Next, after every local graph mended with synthetic neighbors generated by the respective NeighGen model, the system executes FedSage to obtain the generalized node classification model. Algorithm 1 shows the pseudo code for FedSage+. To perform node classification on G, we consider a GNNF with K aggregation operations1 and each aggregation operation contains Rfully-connected layers.