Later in the article it seems like the original stating is more consistent with what you said, but everything I've ever learned in network theory and practice shows that it scales as a power of number of edges, and the logarithm of the number of nodes.
The first is the number of nodes, the second edges.
It could explain my misunderstanding, and also seems consistent with the explanation later in the article, but it's also completely the opposite of what we observe on the internet; for example, the value of the web is definitely not in its in number of pages, but in the value and quality of the connections between the pages.
Metcalfe's law says value increases as 1-3-6-... instead of 2-3-4.
In graph terms, users are nodes, connections are edges, and in a fully-connected graph edges are in order of the square of nodes.
But ethernetworks aren't fully connected (they tend to have lots of local connections that then are connected to each other through routing).
Realistically, the only metric that I can think of that makes sense here isn't proportional to |V| or |E| but to the betweenness connectivity of the graph and the average distance between nodes.