The fundamental idea underpinning the GKAT architecture is the observation that the attention matrix $A_i$ from GAT can be "densified" as follows:

$$A_i(k, l) = \frac{K(q_k, k_l) T(k, l)}{\sum\limits_{r \in V} K(q_k, k_r) T(k, r)} \tag{2}$$

where $K : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is a kernel defined on feature vectors in nodes of the graph and $T : V \times V \rightarrow \mathbb{R}$ is another kernel defined on the nodes of the graph $G$ (that for simplicity will depend only on the topology of the graph, but not feature vectors in those nodes).