The role of the decoder is to reconstruct certain graph statistics from the node
embeddings that are generated by the encoder.

Decoder Framework in Shallow Embedding Approach

The encoder is a function that maps nodes $v \in \mathcal{V}$ to vector embeddings
$ z_v \in {\rm I\!R}^d$
(where $z_v$ corresponds to the embedding for node $v \in \mathcal{V}$).

Encoder Framework in Shallow Embedding Approach

Encoder and decoder graph framework is used for learning node embeddings for graph representation learning.

Encoder maps node in a graph to a low dimensional vector space.

Decoder uses the low dimensional node embeddings to reconstruct the node's position in the original graph.

San Jose State University

Node embeddings is a way of graph representation. The goal of node embeddings is to map the nodes in the graph into a low-dimensional vector space, where the relations of the nodes in graph can be reflected by the geometric relations in the embedding space. Compared with hand-engineered features, node embeddings are more flexible and they are learned through learning processes.

Node 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

Graph Representation Learning by William Hamilton

Summary of Shallow Embedding Algorithms.

Encoder and Decoder Graph Framework 

Matrix factorization is used  to learn a low-dimensional approximation of a node-node similarity matrix S, where S generalizes the adjacency matrix and captures some user defined notion of node-node similarity.

Encoder-Decoder idea based on Matrix
Factorization

Node embeddings are optimized so that two nodes have similar embeddings
if they tend to co-occur on short random walks over the graph.

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