To address this complication, we augment our decoder to make it multirelational. Instead of only taking a pair of node embeddings as input, we now define the decoder as accepting a pair of node embeddings as well as a relation type. We can then interpret the output of this decoder as the likelihood that the edge exists in the graph.

Reconstructing Multi-Relational Data

Embedding multi-relational graphs can be framed as a reconstruction task. Given the embeddings $$z_u$$ and $$z_v$$ of two nodes, the goal is to reconstruct the relationship between these nodes. The primary complication in this task, compared to embedding simple graphs, is the presence of multiple different types of edges that must be accounted for during reconstruction.

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Multi-relational graphs have different types of edges. The edge notation can be rewritten as $$(u, \tau, v) \in \mathcal{E}$$, where $$\tau$$ is the relation type. An adjacency matrix $$A_{\tau}$$ has to be defined for each edge type. The entire graph is summarized by an adjacency tensor $$\mathcal{A} \in \mathbb{R}^{|\mathcal{V}| \times |\mathcal{R}| \times |\mathcal{V}|}$$, where $$\mathcal{R}$$ is the set of relations. Two important subsets of multi-relational graphs are:
- Heterogeneous graphs
- Multiplex graphs

Multi-Relational Graphs

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 this type of graphs the nodes  are inspired by the edge type , set of nodes in the graph are disjoint  $\mathcal{V} = \mathcal{V_{1}} \cup \mathcal{V_{2}} \cup ... \cup \mathcal{V_{k}}$ where $\mathcal{V_i}  \cap \mathcal{V_j} = \emptyset$, $\forall_i \neq j$.

Heterogeneous Graphs

Multiplex graphs can be modeled as $$k$$ layers, where each layer showcases a unique relation and every node belongs to every layer. Inter-layer and intra-layer edges define the connections between nodes across the layers.

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