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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Multi relational graphs have different types of edges.
The edge notation can be rewritten as $(u, \tau,v) \in \mathcal{E}$
$ \tau $ - relation type

Adjacency matrix $ A_{\tau}$, has to be  defined for each edge type.

Entire graph is summarized by an adjacency tensor $ \mathcal{A} \in {\rm I\!R} ^{|\mathcal{V}| \times |\mathcal{R}| \times |\mathcal{V}| }$
$\mathcal{R}$ , 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

We can view embedding multirelational graphs as a reconstruction task. Given the embeddings $z_u$ and $z_v$ of two nodes, our goal is to reconstruct the relationship between these nodes. The complication—compared to the setting of the previous chapter—is that we now have to deal with the presence of multiple different types of edges.

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