## Graph

A graph $\mathcal G$ has nodes $\mathcal V$ and edges $\mathcal E$,

$$\mathcal G = ( \mathcal V, \mathcal E).$$

Edges

Edges are relations between nodes. For $u\in \mathcal V$ and $v\in \mathcal V$, if there is an edge between them, then $(u, v)\in \mathcal E$.

## Representations of Graph

There are different representations of a graph.

A adjacency matrix of a graph represents the nodes using row and column indices and edges using elements of the matrix.

For simple graph, the adjacency matrix is rank two and dimension $\lvert \mathcal V \rvert \times \lvert \mathcal V \rvert$. For edge $(u, v)\in \mathcal E$, it is represented by the matrix element $\mathbf A[u,v]=1$.

See Sanchez-Lengeling, 2021 for an interactive example1.

### Laplacians

Laplacians are transformations of the adjacency matrix but provides a lot more convenience for analysis.

## Multi-Relational Graph

A graph may have different types of edges, $\tau\in \mathcal R$, where $R$ is a set of types of relations. A multi-relational graph is then

\begin{align} u &\in \mathcal V \\ v &\in \mathcal V \\ \tau &\in \mathcal R \\ (u, \tau, v) &\in \mathcal E. \end{align}

For multi-relational graph, it can have more ranks and the dimension can be $\lvert \mathcal V \rvert \times \lvert \mathcal V \rvert \times \lvert \mathcal R\rvert$, where $\mathcal R$ represents the types of relations.

$\tau=\tau_1$ $A_{\tau=\tau_1}$
$\tau=\tau_2$ $A_{\tau=\tau_2}$
$\cdots$ $\cdots$

Two popular examples:

• heterogeneous
• multipartite
• multiplex

### Heterogeneous

Nodes are subsets without intersections, $\mathcal V = \mathcal V_1\cup \mathcal V_2 \cdots \mathcal V_k$ and $\mathcal V_i \cap \mathcal V_j = \emptyset$ for $\forall i\neq j$.

The famous multi-partite graph is an example of heterogeneous graph.

Movie and User Bi-partite Graph

For a website that hosts a movie database and a user database, the relation about which user watched which movie is a bi-partite graph.

### Mutiplex

The edges represent different