Random Search Neural Networks for Efficient and Expressive Graph Learning

Random walk neural networks (RWNNs) have emerged as a promising approach for graph representation learning, leveraging recent advances in sequence models to process random walks. However, under realistic sampling constraints, RWNNs often fail to capture global structure even in small graphs due to incomplete node and edge coverage, limiting their expressivity. To address this, we propose \textit{random search neural networks} (RSNNs), which operate on random searches, each of which guarantees full node coverage. Theoretically, we demonstrate that in sparse graphs, only $O(\log |V|)$ searches are needed to achieve full edge coverage, substantially reducing sampling complexity compared to the $O(|V|)$ walks required by RWNNs (assuming walk lengths scale with graph size). Furthermore, when paired with universal sequence models, RSNNs are universal approximators. We lastly show RSNNs are probabilistically invariant to graph isomorphisms, ensuring their expectation is an isomorphism-invariant graph function. Empirically, RSNNs consistently outperform RWNNs on molecular and protein benchmarks, achieving comparable or superior performance with up to 16$\times$ fewer sampled sequences. Our work bridges theoretical and practical advances in random walk based approaches, offering an efficient and expressive framework for learning on sparse graphs.

Learning Laplacian Positional Encodings for Heterophilous Graphs

In this work, we theoretically demonstrate that current graph positional encodings (PEs) are not beneficial and could potentially hurt performance in tasks involving heterophilous graphs, where nodes that are close tend to have different labels. This limitation is critical as many real-world networks exhibit heterophily, and even highly homophilous graphs can contain local regions of strong heterophily. To address this limitation, we propose Learnable Laplacian Positional Encodings (LLPE), a new PE that leverages the full spectrum of the graph Laplacian, enabling them to capture graph structure on both homophilous and heterophilous graphs. Theoretically, we prove LLPE's ability to approximate a general class of graph distances and demonstrate its generalization properties. Empirically, our evaluation on 12 benchmarks demonstrates that LLPE improves accuracy across a variety of GNNs, including graph transformers, by up to 35% and 14% on synthetic and real-world graphs, respectively. Going forward, our work represents a significant step towards developing PEs that effectively capture complex structures in heterophilous graphs.

Understanding GNNs and Homophily in Dynamic Node Classification

Homophily, as a measure, has been critical to increasing our understanding of graph neural networks (GNNs). However, to date this measure has only been analyzed in the context of static graphs. In our work, we explore homophily in dynamic settings. Focusing on graph convolutional networks (GCNs), we demonstrate theoretically that in dynamic settings, current GCN discriminative performance is characterized by the probability that a node's future label is the same as its neighbors' current labels. Based on this insight, we propose dynamic homophily, a new measure of homophily that applies in the dynamic setting. This new measure correlates with GNN discriminative performance and sheds light on how to potentially design more powerful GNNs for dynamic graphs. Leveraging a variety of dynamic node classification datasets, we demonstrate that popular GNNs are not robust to low dynamic homophily. Going forward, our work represents an important step towards understanding homophily and GNN performance in dynamic node classification.