This isn't a field I'm entirely up on the research in, but it does overlap with my old interest in biochemistry and my current work in archaeology. Some of what I write here may therefore be easily corrected by those in the know, but I had a few problems with the paper (which I enjoyed overall). Please enlighten me in the comments.
Inconsistent Data
Table 1 of the pairwise genetic distances between Minoans and other populations seems to be based on the data presented in Supplementary Table S5, but it doesn't give the same data. For instance, the first group is Bronze Age Sardinians, which have an average pairwise distance of 2.75111 in S5, but 2.89 in Table 1. Maybe that's because these two tables were calculated using two different versions of the same software (Arlequin 3.5.1.2 vs 3.5.1.3), but that appears unlikely, since the Arlequin updates page doesn't seem to list any such change between versions.
[BTW, if you're going to give a supplementary file of data, why not put it in some nice format like CSV instead of a PDF? I fixed it up with Tabula, but that was an unnecessary expenditure of my time.]
"Ancient"?
The authors divide their comparison population groups into two chronological categories, "ancient" and "modern." Their ancient group includes 11 distinct samples: 6 neolithic, 3 Bronze Age, 1 medieval/Iron Age(!), and 1 Byzantine. They seem to be taking the nomenclature of their original sources, so it's worth noting that Byzantine means 11th −13th centuries and Iron Age-Medieval-Nordic means 1st - 14th centuries (and that the original authors did not apply this catch-all term). I didn't look up all their "modern" samples, but the remarkable chronological span of the their "ancient" ones already raises some doubts in my mind. You can't lump together millennia of data like this and act like you have homogeneous groups. This makes map b in Figure 3 misleading as well, even ignoring the way the map is colored in areas where there is no data at all (like all of North Africa, on which more below). Line c of Figure 4 does break these ancient sets up into three distinct groups, leaving out the Neolithic, but even this obscures the fact that one of these groups has a much higher sharing than the others: the one from BA Sardinia with 52%, compared with a maximum of 12% for the other two BA sites, and 15% for all four of the other locations. I'd question this approach, as it's a great example of why the mean can sometimes be a bad choice for comparison: with an N of 3 in the BA group and a standard deviation that's larger than that, you'd be better off reconsidering your choice to group these data points, or at least use the median.
If we break out this Sard group then, we're left with a much more consistent sharing with the Neolithic groups (mean=24%), and fairly low sharing with the other BA, the Byzantine, and the (absurd) Iron Age-medieval group. In short, with that one Sard exception, all the European post-Neolithic "ancient" groups match the Minoans about as well as the modern Middle Eastern/Turkish/Caucasian sets. Since the Minoans are BA Mediterraneans, that makes me wonder why the Minoans are so different...you know, assuming again that that small N isn't to blame.
Line d of that same Figure 4 breaks down the six Neolithic locations into three sub-groups based on a north-south positions. Again, I'm not sure how far you want to go with an N of 1 in the northern set. The largest sharing with the "southern" group—really France and Spain—is not surprising, given their proximity to the Mediterranean and the likely mixing of those populations in the Neolithic (as discussed, for example, in the article they take one of their datasets from).
More troubling though is the complete lack of any ancient sample from anywhere further outside Europe than Asia Minor (modern Turkey). If you're going to claim that Minoans aren't like ancient Egyptians or Middle Easterners, you'd probably want to have some sample of those groups to compare them to. The authors do note the lack of the typical African L haplotype in the Minoan group, but I'd still like to see a direct comparison with some North African Neolithic and BA data.
Note revealing my ignorance
I find the methodology of creating the haplotype sharing quite interesting. Count the individuals from each set who have a haplotype that appears in any of the Minoan individuals, then divide by the total number of individuals in that set to get a percentage sharing. For example, say you have a population with three haplotypes, A, B, and C, distributed in the population at 80%, 10%, and 10%, respectively. Another population with all the same haplotypes distributed 1%, 1%, 98% would have 100% sharing, while a third population with 85% A and 15% B would score 90%, even though the distribution of haplotypes was more similar than the second group's in overall distribution.
So in the data in this paper, we see that the Sard BA sample had the largest overlap of any group, but with only two shared haplotypes, while the North Africa set had a much lower sharing percentage, but 6 shared haplotypes. I realize part of the problem is the small N's involved, which often don't permit for a great representation of the overall distribution of haplotypes within the total population.
Update: A post from a more knowledgeable guy than myself are here.
Update: A post from a more knowledgeable guy than myself are here.