Personally I'd like to see some research on what effect 4mb weight blocks would have on further centralization, but that's what i foresee most likely to happen.
I'm not too sure that much research is necessary.
All "4MB weight" means is that in addition to the standard 1 MB blocks we have now, an extra 3 MB can be used
but for signatures only. Today's transactions dedicate slightly less than half of their size in bytes to the signature, on average.
And that's why the increase gets quoted as only being between 1.75 MB and 2MB, because it's not possible to fill up all 3 MB of extra block space using transactions that are 1:1 data:signature. The reason there's 3x times more space than we can use is that Lightning transactions will change that average ratio closer to the 1:3 data:signature due to it's use of multi-sig (multi-sig transactions must have larger signatures, as more than one signature is required to sign them correctly).
All this would have to change if the signature technology got changed, which is a likely future upgrade. Bitcoin uses ECDSA signatures right now, but more space-efficient signature schemes exist that could replace it, that would mean the 1:3 data:signature ratio would need changing to reflect that.
So, I hope you can see from this that neither node or mining centralisation is affected by the weighting, it's just a ratio used to target efficient use of block space.