The goal of random walk embeddings is to optimize node representations so that the pairwise decoder, defined via a softmax function, approximates the probability of co-occurrence on random walks:

$Dec(z_u, z_v) = \frac{e^{z_u^T z_v}}{\sum_{v_k \in V} e^{z_u^T z_k}} \approx p_R(v | u)$

where $p_R(v | u)$ is the probability of visiting node $v$ on a short random walk starting from node $u$.