Universal scaling across biochemical networks on earth

Following up on this post, some of the same authors in the last reference followed up with this article continuing with the theme that

"while biochemical networks are not scale-free, they nonetheless exhibit common structure across different levels of organization, independent of the projection chosen, suggestive of shared organizing principles across all biochemical networks."

So what then are shared organizing principles and common structure across different levels of organization? They "are characterized by scaling laws" in a "nested hierarchy" but are not 'scale-free.' And yet they are universal in that they "scale in the same way across levels." However they "did not require the scaling coefficients to be exactly the same [...] but we did require the same fit to be shared across our data (indicating the possibility of shared generative mechanisms) to qualify as universal."

So it seems the apparent discrepancy, at least to me, is in what they determine as "the same fit to be shared across our data." What that "same fit" entails is lost to me in all the technical and statistic jargon. Anyone else have any better luck with deciphering this?

It apparently has to do with the following, but I can't make heads or tails of it:

"We therefore focus on topological measures such as average shortest path length, average clustering coefficient, and assortativity (degree correlation coefficient)."