Definition:Network

Definition
A network $N = \left({G, w}\right)$ is:
 * a graph or digraph $G = \left({V, E}\right)$

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

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

Also known as
A network $N = \left({\left({V, E}\right), w}\right)$ 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 $f \left({u v}\right)$ 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.