Network with Positive Integer Mapping is Multigraph

Theorem
Let $N = \struct {V, E, w}$ be a network whose weights are all strictly positive integers.

Then $N$ can be represented as a multigraph.

Conversely, any multigraph can be expressed as a network whose weights are all strictly positive integers.

Network as Multigraph
Let $N = \struct {V, E, w}$ be a network, either directed or undirected.

, suppose $N$ is undirected, and in the following argument allow the term edge to include arcs.

Let $e \in E$ be an edge of the underlying graph $\struct {V, E}$ of $N$.

Let $e = u v$, where $u, v \in V$ are vertices of the underlying graph $\struct {V, E}$ of $N$.

Let $n = \map w e$ be the weight of $e$.

By hypothesis, $n$ is a strictly positive integer.

Let $n$ edges be identified between $u$ and $v$.

Repeat this process for all edges of $N$.

The resulting object is a multigraph.

Multigraph as Network
Let $G = \struct {V, E}$ be a multigraph, where each element of $E$ may occur multiple times.

Let $e = u v$ where $e \in E$ and $u, v \in V$.

Let $\map m e$ be the multiplicity of $e$.

Then $m$ can be identified with a weight function from $w: E \to \Z_{>0}$.

Thus the network $N = \struct {V, E, w}$ can be created.