Definition:Network

Definition
A network is a graph or digraph together with a mapping which maps the edge set into the set $$\R$$ of real numbers.

It can be depicted by writing the numbers next to the edges of the graphs involved.

A network is also known as a weighted graph, the numbers that each edge is assigned to being called the weights.

Directed Network
A directed network is a network resulting from a digraph:


 * DigraphNetwork.png

Undirected Network
An undirected network is a network resulting from a simple 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 is allowed to have loops.

Network as Multigraph
An undirected network whose mapping is into the set $$\Z_+$$ 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 $$f \left({u v}\right)$$ edges between each vertex $$u$$ and $$v$$.

Compare with
It can be seen that an undirected network can be considered as an edge-colored graph in which the colors are each assigned numbers.