Understanding GNNs and Homophily in Dynamic Node Classification

TL;DR

This work proposes dynamic homophily, a new measure of homophily that applies in the dynamic setting that correlates with GNN discriminative performance and sheds light on how to potentially design more powerful GNNs for dynamic graphs.

Abstract

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.

Figures (10)

• Figure 1: Toy dynamic graph where the task is to predict future node labels. When $t=0$, GNNs obtain good performance in predicting future node labels. Dynamic homophily is high since all nodes have the same future label as their neighbors' current label while, static homophily is low since many nodes (e, g, h, i) and their neighbors have different labels at the current timestep. When $t=2$, GNNs obtain poor performance. Dynamic homophily is low since many nodes (a, b, c, d) have a different future label than their neighbors' current label, while static homophily is high since all nodes and their neighbors have the same current label at $t=2$.
• Figure 2: Expected AUROC across GCN layers as a function of dynamic homophily levels. The AUROC of odd layer GCNs is monotonically increasing in dynamic homophily levels, while the AUROC of even layer GCNs increase as both dynamic homophily levels approach 0 and 1, providing potential insights into how to design more powerful dynamic GNNs that can adapt to dynamic homophily levels across time.
• Figure 3: Mean $\pm$ standard deviations of Spearman's rank correlation coefficient between GNN AUROC and homophily measures across graphs in the test set for a subset of datasets. For most GNN and dynamic graph combinations, dynamic homophily has a higher correlation with GNN performance compared to static homophily.
• Figure 4: Mean and standard deviations of dynamic homophily, static homophily, and AUROC across all graphs in the test set for the Math, Higgs 3, and Gavin dynamic graph. For all three dynamic graphs, static homophily stays high across time, while dynamic homophily and GNN performances exhibit the same trends. We do not connect AUROCs across signal spreading tasks since we only predict for nodes which the signal has not reached.
• Figure 5: Mean $\pm$ standard deviations of Spearman's rank correlation coefficient between GNN AUROC and homophily measures across graphs in the test set for the remaining dynamic graph datasets. For most GNN and dynamic graph combinations, dynamic homophily has a higher correlation with GNN performance compared to static homophily.

Theorems & Definitions (20)

• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Definition 3.5: Dynamic homophily
• Lemma B.1
• proof
• Theorem B.2
• proof
• Theorem B.3