The noise contrastive approximation approach of Node2Vec approximates the loss function as follows:

$$L = \sum_{u,v \in D} -\log(\sigma (z_u^T z_v)) - \gamma E_{v_n \sim P_n(v)}[\log(-\sigma (z_u^T z_{v_n}))]$$

University of Michigan - Ann Arbor

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Random walk embedding methods often minimize the following cross-entropy loss: $$L = \sum_{u,v \in D} -\log(DEC(z_u,z_v))$$

Cross-Entropy Loss in Random Walk Embeddings

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

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