When are two networks the same? A single network has many representations, because the position of the vertices and the lengths of the edges are free to be changed. Play around with these applets to see the range of representations a network (both directed and undirected) can have:
As you’re moving the vertices around, think about the kinds of changes we can make, and the kinds of changes we can’t. Here are the most important things you may have already noticed:
- If two vertices start connected they stay connected.
- In directed graphs, arrows don’t reverse direction.
- If two vertices don’t start connected they stay disconnected.
- We don’t add or delete vertices or edges.
As long as we stick with those rules, moving the vertices around just like in the applets, we are only changing the representation of the network, not the network itself.
Still, it can be hard to tell when several networks that look very different are actually just different representations of the same network (sometimes we then say the networks are isomorphic). These are four different representations of the Clebsch network:
You can pick one representation, move the vertices around, and turn it into any of the other three. Try it out for yourself with the applet below:
Can you find the other three?