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Well, the interesting thing isn't that such graphs exist, so finding an arbitrary graph with this property isn't really relevant.

It's that certain distributions of random graphs, when sampled tend to produce graphs with this property and human social networks are one of them.

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No, this statement is true for every graph that doesn't have constant degree. It's not really to do with random graphs. Rather it's the fact that a random neighbor of a random node draws from a different distribution than a random node, and the former distribution puts more mass on nodes of higher degree.

Ah, oops, you are correct.

True. My comment was to highlight how wrong my initial intuition was.

Looking around, this article[1] by Arun Maiya and Tanya Berger-Wolf explores this in the context of social networks. Fascinating stuff!

[1] https://arxiv.org/pdf/1109.3911.pdf

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