Some people have difficulty staying excited by the sheer beauty of a mathematical construct. They need to find some practical use. As it happens, the kind of thinking illustrated in this presentation does have practical applications in addition to pure beauty. Here are a few examples:
The book by Arthur Loeb which inspired this presentation was itself inspired in part by conversations about crystal structures. For this purpose, the nodes are atoms and the relationships are the forces which hold the atoms together. The mathematics shows which arrangements of the atoms are possible and which ones are more stable.
In studying living communities, the nodes are populations are the relationships may be nutrient flows: Hawks eat mice who eat seeds of plants which absorb nutrients in the soil which microbes make available by breaking down the waste from the hawks. And that defines a feedback loop (which is a "shape" to us).
In such applications, edges are often made directional (we eat corn but corn does not eat us) and assigned weights based on how many calories or other nutrient value is transferred. The resulting geometry can be quite a bit more complex than what we have been looking at.
If the nodes are electronic devices and the relationships are communication links, this kind of geometry can be used to describe networks and networks of networks such as the internet. Or the nodes can be documents and the relationships can be links; then we can describe the World Wide Web.
Many networks are not connected structures of the type we emphasized but instead have "root" and "leaf" nodes that only connect on one side. However, very large networks such as the internet will be well connected on the interior; the "leaf" nodes are your phones and tablets. Very small networks can form structures very much like the ones we talked about.
This type of mathematics can also be used to study human relationships. People are the nodes, while the relationships might be friend-to-friend or parent-child or employer-employee. Or the nodes could be larger groups like commercial companies or governmental entities.