# What is Graph

## Graph

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

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

## Representations of Graph

There are different representations of a graph.

### Adjacency Matrix

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 example

^{1}.

### Laplacians

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

Laplacian is a useful representation of graphs. The unnormalized Laplacian is

\mathbf L = \mathbf D - \mathbf A,

where $\mathbf A$ is the and $\mathbf D$ is the degree matrix, i.e., a diagonalized matrix with the diagonal elements being the degrees.

Normalized Laplacian

The symmetric normalized Laplacian is

\mathbf L_{\text{sym}} = \mathbf D^{-1/2} \mathbf A \mathbf D^{-1/2}.

The eigenvalues of normalized Laplacian is bounded ($$).

The random walk Laplacian is

\mathbf L_{\text{RW}} = \mathbf …

## 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

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

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.

### Mutiplex

The nodes can be viewed to exist in different layers and the edges in each layer are different.

- Hamilton2020 Hamilton WL. Graph Representation Learning. Morgan & Claypool Publishers; 2020. pp. 1–159. doi:10.2200/S01045ED1V01Y202009AIM046
- Sanchez-Lengeling2021 Sanchez-Lengeling B, Reif E, Pearce A, Wiltschko AB. A Gentle Introduction to Graph Neural Networks. Distill. 2021;6. doi:10.23915/distill.00033

L Ma (2021). 'What is Graph', Datumorphism, 09 April. Available at: https://datumorphism.leima.is/wiki/graph/basics/what-is-graph/.