Definition:Network/Weight

This page is about weight in the context of network theory. For other uses, see weight.

Definition

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 $N$.

Let $PQ = v_1 e_1 v_2 e_2 \ldots e_{n - 1} V_n$ be a walk in $N$.

The weight of $PQ$ is defined as the sum of the weights of the edges that constitute $PQ$:

$\map w {PQ} := \ds \sum_{k \mathop = 1}^{n - 1} \map w {e_k}$

This definition can also be applied to paths, trails, circuits and cycles.

In the context of network analysis, the weight of an edge of a network is often referred to as a penalty.

Results about weight in the context of network theory can be found here.