Yes! It is indeed a graph of the lightning network. Each lightning circle represents a node, and each line from one node to another represents a payment channel.

There are many different ways to draw this same graph, but with the approach I took I think it should minimize the distance between connected nodes.

The more connected you are the closer you are to the center of this graph.

Information about nodes is relayed throughout the network via a gossip protocol. So as long as there is at least one main connection to the network. The information from a distant node eventually makes it to all nodes on the network.

My understanding is it does this by basically by passing messages to and from your peers and then to/from your peer's peers and so on.

That's an interesting idea. It might be possible to expand this implementation to drawing graph generically. However even if I generalized this software, drawing something like the internet might have additional obstacles. With the internet there is a lot of private information that makes it difficult to do this same thing I have done for lightning. All the information collected here is publicly relayed throughout the lightning network. So you would probably have to be an Internet Service Provider to obtain information about the connection data for the internet.