Definition:Network
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.
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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Node
The vertices of a network are usually referred to as its nodes.
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
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- 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): network
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): weighted graph
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): network
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): weighted graph
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): weighted graph