# “Two-scale” network?

I've read that networks can be:

• random (Erdős–Rényi model),

• scale-free (Albert–Barabási model),

• small-world (Watts–Strogatz model).

But can a real world network be “two-scale”, in the sense that its degree distribution only consists of two different degrees, for example $$(5,5,5,5,4,4,4,...)$$ where the number nodes of degree $$4$$ is equal to $$9$$?

• Real world, no. But for a random network as you wish, yes, simply adapt the usual construction starting with half-edges attached to each vertex. – Did Aug 28 '18 at 5:57
• It can't be real world because it doesn't have a power law degree distribution? Would the degree distribution for a "two scale" network just be a line segment with negative slope? – Ultradark Aug 28 '18 at 13:17
• It can't be real world because it doesn't have a power law degree distribution? – Many real networks do not have such a distribution and it is still up to debate to what extents observations of such distributions in real networks are a measurement artefact. The models you list are just that: models. Any given real network will substantially differ from the typical output of these models. (And note that I use typical here only because all of the models involve randomness and therefore can produce all sorts of things with a very low probability.) – Wrzlprmft Jan 14 at 18:25

If you are asking whether there exists a graph that has that particular degree sequence, then the answer is yes, and here is an example given by its edge list: $$[(0, 11), \; (0, 3), \; (0, 3), \; (0, 10), \; (0, 6), \; (1, 4), \; (1, 3), \; (1, 11), \; (1, 10), \; (1, 5), \; (2, 5), \; (2, 7), \; (2, 7), \; (2, 4), \; (2, 3), \; (3, 5), \; (4, 8), \; (4, 8), \; (5, 6), \; (6, 9), \; (6, 11), \; (7, 7), \; (8, 12), \; (8, 10), \; (9, 10), \; (9, 12), \; (9, 11), \; (12, 12)]$$
Also note that networks can fall in much more than just these three categories, and the Watts–Strogatz model with $$0$$ randomness actually does not exhibit the small-world property.