Six degrees of separation refers to shortest paths, and fast mixing is decidedly _not_ the intuition behind it.
[1] https://www.comp.nus.edu.sg/~yuhf/yuh-sybillimit.pdf
[2] http://www.internetsociety.org/sites/default/files/dane.pdf
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EDIT: Wow, I've read that paper you linked to. So they use the data-set from SNAP like Facebook A and Facebook B.
But, if I understand correctly, these data-sets were formed by combining "egonets", which means they took a set of people and a small ball around them, and put all those balls in the graph. That explains the horrendous eigenvalues.
In fact, the main problem with the paper is the opposite -- the Facebook graphs they use are regional networks borrowed from [1]. This means that Facebook's actual mixing time should be _dramatically higher_ than the times measured in the paper because of the tendency of random walks to get stuck in regional networks. I believe this is the reason their Facebook-A and Facebook-B mixing times are much lower than the others, such as LiveJournal.
Alvisi et al. have a couple of related papers specifically looking at the implications of our new understanding of social-graph random walks for sybil defenses. [2, 3]
[1] https://www.cs.ucsb.edu/~ravenben/publications/pdf/interacti...
[2] http://www.cs.utexas.edu/users/lorenzo/papers/Alvisi13SoK.pd...
[3] http://www.cs.utexas.edu/users/lorenzo/papers/Alvisi14Commun...
You're right, 6 degrees of separation is a statement about diameter, but a small diameter and mixing speed are related.