[zvr]

Two users in the network: A and B; one connection: AB. Three users in the network: A, B, and C; three connections: AB, AC, BC. Four users in the network: A, B, C, and D; six connections AB, AC, AD, BC, BD, CD.

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.

[dekhn]

Yes, I see I had completely misread and misunderstood the original law.

But ethernetworks aren't fully connected (they tend to have lots of local connections that then are connected to each other through routing).

[jacquesm]

I think the difference between logical and physical connections is what drives the confusion here. If two nodes can reach each other somehow then for Metcalfe's law they are connected, even if there is no direct connection between them.

[dekhn]

Yes, I realized that shortly after reading the replies. Thanks for stating it explicitly. Once again, my brain's inability to parse english caused a multi-decade misunderstanding.

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.

[jacquesm]

You actually have a very valid point: given that there is such a thing as the maximum ttl at some point that 'logically connected' network will become more and more sparse depending on how 'wide' the network really is. I wonder if there are already parts of the V4 net that are so far removed from each other that this is an issue.