Question

Consider undirected networks without self-loops or duplicate links. Given N nodes and a

variable number of m edges, for what value of m do we obtain the largest diversity of networks? And for

this m, how does the number of networks scale with N?

Answer 1

We have N nodes in the network.

Therefore the maximum number of edges in the network = = N(N-1)/2which occurs when we have edges between every 2 nodes in the graph.

Since we have m total edges, we can have any m out of these N(N-1)/2 edges.

Hence the number of networks = .

This is the maximum for m = N(N-1)/4

For m = N(N-1)/4.

The number of networks scales with N as:

The number of networks = which increases as N increases.

You can comment below the answer in case of any doubts and I will be happy to help.

Please give a thumbs up if the answer could be of help!

All the best!