Understanding Virtual Nodes: Oversmoothing, Oversquashing, and Node Heterogeneity

Message passing neural networks (MPNNs) have been shown to have limitations in terms of expressivity and modeling long-range interactions. Augmenting MPNNs with a virtual node (VN) removes the locality constraint of the layer aggregation and has been found to improve performance on a range of benchmarks. We provide a comprehensive theoretical analysis of the role of VNs and benefits thereof, through the lenses of oversmoothing, oversquashing, and sensitivity analysis. First, in contrast to prior belief, we find that VNs typically avoid replicating anti-smoothing approaches to maintain expressive power. Second, we characterize, precisely, how the improvement afforded by VNs on the mixing abilities of the network and hence in mitigating oversquashing, depends on the underlying topology. Finally, we highlight that, unlike Graph-Transformers (GT), classical instantiations of the VN are often constrained to assign uniform importance to different nodes. Consequently, we propose a variant of VN with the same computational complexity, which can have different sensitivity to nodes based on the graph structure. We show that this is an extremely effective and computationally efficient baseline on graph-level tasks.