A heterogeneous graph is a type of multi-relational graph where nodes belong to different types. The overall set of nodes $$\mathcal{V}$$ is partitioned into disjoint sets based on node type: $$\mathcal{V} = \mathcal{V_{1}} \cup \mathcal{V_{2}} \cup \dots \cup \mathcal{V_{k}}$$, where $$\mathcal{V_i} \cap \mathcal{V_j} = \emptyset$$ for all $$i \neq j$$.

San Jose State University

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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

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.

Multiplex Graphs

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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