Networks

New Zealand

Level 7 - NCEA Level 2

Lesson

Edges can be either **directed **or **undirected**. Directed edges (sometimes called **arcs**) are represented by arrows and symbolise a one-way relationship, while undirected edges are represented by lines and symbolise a two-way connection.

If there are directed edges in the network, we call the network a **directed network** (or **digraph**). If there are no directed edges in a network, we call the network an **undirected network**. Look back through the networks you’ve seen so far - the ones that have arrows are directed, and the rest are undirected.

**Q: What about if there's a mix - some arrowheads, some lines?**

A: Edges in undirected networks represent two-way connections. If there are directed edges in a network, then one-way connections are possible - so we turn the lines representing two-way connections into two arrows, like this:

*We can still express a two-way connection using directed edges, we just have to draw in an extra (directed) edge. You typically don't see "mixed" networks - it just makes things easier if either all the edges are lines, or all the edges are arrows.*

An edge that **starts and ends at the same vertex** is called a **loop**. A network that has

- no loops, and
- no two vertices connected by more than one edge

is called a **simple network**. Most networks we will see will be simple.

Here’s a quick summary of the definitions we’ve seen so far in this lesson and the previous one.

Summary

**Vertex **- A circle or dot in a network. Often given a **vertex label**.

**Edge **- A line segment connecting a vertex to a vertex. Can be **directed **(arrowhead) or **undirected **(no arrowhead). If it starts and ends at the same vertex, we call it a **loop**.

**Network **- A collection of vertices and edges. Can be **directed **(arrowheads) or **undirected **(no arrowheads). If there are no loops and no “repeat” edges, the network is **simple**.

Choose appropriate networks to find optimal solutions

Apply network methods in solving problems