Escaping Oversquashing: Addressable and Support-Aware Global Memory for Message Passing Networks

Virtual nodes are a natural tool against oversquashing: they replace long message-passing paths by a two-hop global route. But when many nodes share one global state, that shortcut can become a bottleneck itself. We study two properties of this global memory. First, addressability: under constant-margin address codes and a nonlinearity that amplifies this margin, multiplicative write/read maps provide $M$ selectable memory rows with only $O(\log M)$ address-code dimensions. Cross-attention slots and a constrained $ELU+1$ bilinear memory both satisfy these conditions. Second, support awareness: normalized cross-attention has no self-key for a latent query to use as a reference. A learned private anchor supplies this reference, keeps the read bounded, and exposes the strength of the matching source mass. We demonstrate the merits of such properties on several instances of Two-Radius and Tree-NeighborsMatch: both addressable realizations solve the controlled tasks through depth $5$, where pooled VNs of comparable or larger size reach about $10.6$.