Definition:Network
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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:
Undirected Network
An undirected network is a network resulting from a simple 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 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.
Sources
- Gary Chartrand: Introductory Graph Theory (1977): $\S 1.6$
