[dekhn]

So, the wikipedia first line is wrong, you're saying? "Metcalfe's law states that the financial value or influence of a telecommunications network is proportional to the square of the number of connected users of the system"

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.

[gnramires]

The number of connected users is different from the number of user connections :)

The first is the number of nodes, the second edges.

[dekhn]

Uhh, are you sure? I believe "connected users" refers to edges. Otherwise it would be stated as "users connected to the network".

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.

[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.