The decoder takes a latent representation Z as input and uses this input to specify a conditional distribution over graphs. 

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In more formal and mathematical detail, to build a VAE for graphs we must
specify some key components.

Key Components of the VAE model

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

Graph Representation Learning by William Hamilton

 In the case of graphs, the probabilistic encoder model takes a graph G as input. From this input, qφ then defines a distribution qφ(Z|G) over latent representations. We need to specify the latent conditional distribution as Z ∼ N (µφ(G), σ(φ(G)), where µφ and σφ are neural networks that generate the mean and variance parameters for a normal distribution, from which we sample latent embeddings Z.

A Probabilistic Encoder Model qφ

A Probabilistic Decoder Model pθ

A prior distribution p(Z) over the latent space. We will adopt the standard Gaussian prior Z ∼ N (0, 1), which is commonly used for VAEs.

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