Definition:Network

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Definition

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

a graph or digraph $G = \struct {V, E}$

together with:

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



This article is complete as far as it goes, but it could do with expansion.
In particular: In order to model such things as electric circuits, we need the codomain of $w$ to be more general than $\R$ -- in general a differential equation that defines the behaviour of the component modelled by the edge in question, for example. Out of my depth at the moment. I need to study my Circuits, Devices and Systems.
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Node

The vertices of a network are usually referred to as its nodes.


Weight Function

Let $N = \struct {V, E, w}$ 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:


DigraphNetwork.png


Undirected Network

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

Network.png


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

  • 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

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