We call a network on $n$n vertices complete if every vertex is connected to every other vertex. There is exactly one complete network for each value of $n$n.
The degree of each individual vertex is equal to one less than the number of vertices overall. In other words, if you take $n$n vertices and connect one to all the others, you draw $n-1$n−1 edges from each vertex.
If we take a network and delete some edges, or some of its vertices (and all edges connected to it), we obtain a subnetwork of the original. We often say that one network is a subnetwork of another network if we can get from one to the other through these deletions. For example, any simple network that has $n$n vertices is a subnetwork of the complete network with $n$n vertices - we can just add or delete edges to get from one to the other:
The network on the left has the red edges added in, and then the vertices moved a little. We end up with a complete network. We can then go backwards - start at the right, delete the red edges, and move the vertices a little to recover the original network.
For an undirected network, we call it connected if we can move from any vertex to any other vertex, and disconnected if we can’t. We use the same words for a directed network, though we allow ourselves to move along the (directed) edges in both directions.
The top two networks are connected, the bottom two are disconnected.
These ideas comes up frequently in chemistry, as chemicals are frequently represented as a network. Here are three examples:
The degree of the vertex corresponds to the element. Hydrogen (orange) always has degree $1$1, oxygen (blue) degree $2$2, nitrogen (purple) degree $3$3, and carbon (green) degree $4$4.
These three chemicals have a common subnetwork - the “benzene ring” of six carbon atoms, with some hydrogens still attached:
Each of the three chemical’s networks have a single edge connecting this network to the top part, called the "functional group". Nature, and human chemistry, routinely takes a functional group from a molecule and replaces it with another by deleting and then restoring the edge - you can investigate these biochemical pathways (represented as a network) further here.
