Many large datasets contain data that describes the relationship, or connection, between two or more entities contained within the data set, and many tools are now available for plotting graphs and network diagrams when presented with data that has been structures in a suitable manner.
(A graph is a mathematical structure that can be used to describe these connections in a formal, and easily represented way. In a graph, 'nodes' are connected to each other by 'edges' in either a directed way (a link that goes from one node to another, but not vice versa, such as "A is the parent of B') or an undirected way (the relationship is 'symmetrical' - the M1 motorway connects Leeds to London, and equally connects London to Leeds).)
For example, the IBM Many Eyes visualisation toolkit has a network diagram visualisation that will plot when presented with a set of paired data elements.
Read more detail about the Many Eyes network diagram visualisation, in particular paying attention to how the data must be formatted to work with the visualisation. Create your own dummy data set - or find some data elsewhere that can be suitably arranged - and visualise it using the Many Eyes network diagram visualisation. Does the 'emergent' visualised structure tell you anything interesting about the data?
The length of the connections (edges) between the nodes is not necessarily indicative of anything, and may just be an artefact of the algorithm that lays out the graph.
Here is a map of the internet, circa 2003 that shows the connections between different internet routers.
Watching an animated visulisation of internet maps over time show just how quickly the internet is growing: Cheswick/Burch Map of the Internet, created as part of the Internet Mapping Project (Bell Labs/Lumeta Corporation).
Maps like this can be crudely constructed (in a time consuming way) by using tools such as traceroute that track the routes taken by data packets as they travel between internet addresses.
(There are many traceroute tools available, mostly as downloadable applications that will track, and plot the route taken by) data packets from your computer to other internet connected devices. Some online services will also display a traceroute map of the route followed by data from your computer to their service, and their service to another internet address. If you would like to see an example of this, try using a service such as Visual Trace Route or mapulator.)
(For a historical view of internet connectivity visulisations, see the now lapsed Cybergeography atlas.)