What is Graph

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.

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 example¹.

Graph Adjacency Matrix

A graph $\mathcal G$ can be represented with an adjacency matrix $\mathbf A$. There are some nice and clear examples on wikipedia1, for example, $$ \begin{pmatrix} 2 & 1 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 \end{pmatrix} $$ for the graph Public Domain, Link

Laplacians

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

Graph Laplacians

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

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

Adjacency Matrix for Multi-relation Graph

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.

type	adjacency matrix
$\tau=\tau_1$	$A_{\tau=\tau_1}$
$\tau=\tau_2$	$A_{\tau=\tau_2}$
$\cdots$	$\cdots$

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 $\forall i\neq j$.