Most papers that test for the existence of power laws don't test goodness-of-fit for other distribtions. The lognormal distribution is often just as good a fit to the data. This paper covers a lot of the frequent problems in academic literature that tries to fit power laws:
Preferential attachment is such a beautiful theory because it gives power law distributions of node degree. But real world networks seem to have systematic deviations from power law so often one wonders why more work wasn’t done to find schemes that generate, e.g. lognormal distributions.
It’s possible that preferential attachment isn’t even a good theory for the underlying principle, it’s just that the underlying principle gives fat tails in degree distribution and power laws give okay fat tails.
Sometimes, though, one doesn’t care if it’s a power law per se or just that it has fat tails. In that case why use a power law and not just a better fitting lognormal (or say kernel density estimation)? But power law seems sexy because of its importance in physics (eg scale free, renormalization stuff), so people ran with that when network literature blew up in the mid 2000s.
[1] https://arxiv.org/abs/1801.03400
[2] https://www.quantamagazine.org/scant-evidence-of-power-laws-...