# Definition:Network

## Contents

## Definition

A **network** $N = \struct {G, w}$ is:

together with:

- a mapping $w: E \to \R$ from the edge set $E$ of $G$ into the set $\R$ of real numbers.

### Weight Function

Let $N = \left({V, E, w}\right)$ be a network.

The mapping $w: E \to \R$ is known as the **weight function** of $N$.

A general **network** can be denoted $N = \struct {V, E, w}$ where the elements are understood to be expressed in the order: vertex set, edge set, weight function.

### Weight

Let $N = \struct {V, E, w}$ be a network with weight function $w: E \to \R$.

The values of the elements of $E$ under $w$ are known as the **weights** of the edges of $N$.

The **weights** of a network $N$ can be depicted by writing the appropriate numbers next to the edges of the underlying graph of $N$.

## Directed Network

A **directed network** is a network whose underlying graph is a digraph:

## Undirected Network

An **undirected network** is a network whose underlying graph is an undirected graph:

## Loop-Network

A **loop-network** (directed or undirected) is a loop-graph together with a mapping which maps the edge set into the set $\R$ of real numbers.

That is, it is a **network** which may have loops.

## Also known as

A **network** $N = \struct {\struct {V, E}, w}$ is also known in some contexts as a **weighted graph**.

Hence the elements of the codomain of $w$ are known as the **weights** of the elements of $E$.

Whether **network** or **weighted graph** is used usually depends on the particular application of network theory or graph theory under discussion.

## Also see

- Network with Positive Integer Mapping is Multigraph: An undirected network whose mapping is into the set $\Z_{\ge 0}$ of positive integers can be represented as a multigraph.

Let $f$ be the associated mapping from the edge set $E$ to $\Z_+$.

Then let $u v$ be an edge in $E$.

We create a graph by drawing $\map f {u v}$ edges between each vertex $u$ and $v$.

- Undirected Network as Graph with Edge Coloring: It can be seen that an undirected network can be considered as an edge-colored graph in which the colors are each assigned numbers.

- Results about
**networks**can be found here.

## Linguistic Note

The word **network** is used in the context of a **graph** with a **weight function** because of its connection with the concept in the context of electronics.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Chapter $1$: Mathematical Models: $\S 1.6$: Networks as Mathematical Models - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**weighted graph** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**weighted graph**