Each of these networks is at the lowest level just a collection of nodes with certain connections. But the point is that the overall pattern of these connections is such that on a large scale there emerges a clear correspondence to ordinary space of a particular dimension.
The pictures above are drawn so as to make this correspondence obvious. But what if one was just presented with the raw pattern of connections for some network? How could one see whether the network could correspond to ordinary space of a particular dimension?
The pictures below illustrate the main difficulty: given only its pattern of connections, a particular network can be laid out in many completely different ways, most of which tell one very little about its potential correspondence with ordinary space.
So how then can one proceed? The fundamental idea is to look at properties of networks that can both readily be deduced from their pattern of connections and can also be identified, at least in some